Weeeeell, since you wanted to spark some debate Iāll throw in some views to get the ball rolling! o.o
I do have to note and strongly stress that these answers are purely my personal views and should not be taken as specific directions on how to do things! 
Who is my employer
While it is true that there are only 3 playable factions in the game and we are forced to choose one of them, the nature of the game itself leaves us with plenty of leeway to expand beyond them and still remain canon. I mean really, most of the storyline can only be canon so far. We canāt all have fixed/broken prisons etc. 
Besides, we know that there are a whole slew of other factions in pretty much all the areas we head into so belonging to one of them is canonically not really a problem.
One could argue that the faction lock-in could depend on what you are as well. To go by canon we are all Bees but plenty donāt follow that, so factional allegiance being all over the place would be a logical outcome as well. Peopleās loyalties tend to reflect how deep in an organization they are and humans are very fickle beings after allā¦ Even as newBees allegiances can be pretty shallow.
We also know that the āBig Threeā are not the only factions that have Bees so roleplaying as a member of a faction not in the Big Three should still be perfectly within the scope of the canon even if you do follow the main character canon of the game.
For the easiest reasoning if one does want to be within the Big Three though - You are a āinsert whatever other faction hereā plant within the Templars/Illuminati/Dragon, just to list an example.
What is my job?
Most faction members most likely do have a ānormalā life when they are not busy getting killed or doing said killing.
People just vanishing off the face off the earth, while normally a frequent occurence in the world, would get noticed after a while if it were to happen on a very massive scale and it would be in every factionās interrests for their agents to keep up appearances towards the rest of the world and any prying eyes from other factions.
So yes, you would most likely have a ānormalā job. Going by the canon here, us making our apartment go kaboom and all, we might be relocated and either told to find a job or be given one, to keep up appearances. (Will revisit this bit in later portion)
If anything Iād say the Illuminati might be more likely to assign one to you since they seem very big on the whole control thing, while the other two would probably be either/or when it comes to assigning or letting you pick a job.
Granted, canonically we go from one hotzone to the next with barely any breaks and as such we wouldnāt really have the time for a 9-5 job but lets face it, with content releasing at the rate it is the timeline for pretty much every character is prettyā¦ extendedā¦ So you fill in the ātimeā inbetween with other stuff. Character timelines and Canon timelines will never match up once youāve played for a while so you just have to mix it well, being obscure about timelines and specific dates are a common method.
Social life
To be fair when it comes to a social life we kinda get into a tricky subject.
Lets face it, even if the factions were to tell you no, humans will want to do it anyway.
They are social creatures so they might grow unstable if they were to be denied such things.
Factions would not prevent you from having a social life, although they would probably monitor any relations you might have and be vary of any agent withing their own organization that fraternize with members of other factions. In some cases they might encourage it though, information gathering and the spreading of misinformation is important!
So a social life would probably be encouraged to keep agents happy but at the same time it would be monitored most likely.
With that saidā¦ I have to point out a bit of a pet peeve of mine hereā¦
A lot of RPers have a severe tendency of blurting out their faction alignment at the drop of a hat.
Which is, canonically, a rather quick way to get disposed of. Iām sorry but that is just how it is. The factions are at war with eachother, perhaps not outright sure but there is still a war. Information is everything. Blurting out that you are a āinsert faction hereā is the equivalent of painting a bullseye on yourself and then holding up a 40ft neon sign flashing āI AM HERE!ā. If it is for misinformation purposes or a trap that is one thing but reallyā¦ 99% of the time it isnāt.
Yes, a lot of new Bees or new recruits would not neccesarily adhere to the animosity that the factions have for eachother but precisely because they are new they are also a loooot more expendable for their respective factions. The Secret World is exactly that, secret. And so are the factions that exist within them. You donāt just go around handing out business cards to eachother with your adress listed as āTemplar HQ / The Labyrinth / The Coilā ya know.
Wanting to knock back a few brews with some buddies at the Horned God is all well and good but given the world being what it is, in general youād want to at least hide your colors. At the very least until you know the other patrons can be trusted or at least wonāt knock you out and let you wake up in an interrogation cellā¦ 
Oooook, rant over, sorry about that, moooving on.
When it comes to familiesā¦ they would be a double-edged sword to any faction. On the one hand itās leverage to keep you in line but on the other hand it also makes you a vulnerable asset.
While the above is true the factions would probably not prevent you from having a family, it wouldnāt be feasable to try and prevent such a thing from happening since that would probably create a lot of animosity and friction between an agent and their faction but hey, they could always try to sneak in some chemicals to sterilize you! (Donāt drink the water in the water coolers at the office, 'kay? Or the Kool-Aidā¦)
As to if you should make a familyā¦ ehhh, that would have to be left up to each individual character really. You never know what might happen in life after all.
Oh and yes, Bees are able to have children (But not transfer the Buzzer they have inside them to said child). This has been confirmed by devs on the old forums.
Men in Black
Well, the why is rather straight forward when it comes to Gaia, she needed Bees, quite simple.
As for factions itās even more simple - A bunch of superpowered beings are popping up all over the world, itās a race to snag as many of them as possible to make sure none of the other factions get an advantage that could tip the scales of power in their favor.
This does go for all the factions although their focus might be differing slightly between each of them, but in the end it all boils down to what was mentioned above. Saving the world is all well and good but what is the point in that if you donāt wind up on top afterwards, right?
Keeping in touch with your old life can be risky but I suppose it depends on what the faction decides to do with you. If they want you to maintain your cover they might leave you to pick up your old life, it really does depend on how you make up your character to be honest. For the factions it would be risk versus gain, not to mention if letting you do so would serve their purposes.
As for the rest of the questions on this portionā¦ that all really boils down to whatever you decide on as a backstory, how cooperative you are towards the faction that ārecruitsā you (Seizing all assets you have can be one way to persuade you to complyā¦) and most of it was also touched upon in the previous answers.
Where do we live?
Teeeechnically we donāt actually fully destroy our apartment so one could say that we still live thereā¦ if we were to be sent back to keep up appearances of course.
Logically all of the factions would have safe houses and various assets they could allow you to utilize (Apartments, houses, boats, trailer parks etc. etc.), which seems more likely since they could keep closer tabs on you then.
Besides that itās really up to whatever you prefer.
Personally most of my characters tend to have their own place somewhere in the world but very few of them actually have places that their respective factions do not know of.
Powers, personality, psychology and physiology
We donāt really know much about any ārequirementsā for becoming a Bee, granted there is a lore paragraph where a ācompatible candidateā is described as having ālatent occult abilitiesā, although itās not specifically said that it is a requirement. (A lot of RPers tend to see it that way though lately.) Nor is it the Buzzing that are making those statements specifically, they are simply telling us what a group of people said for a certain experiment involving implanting one of the Buzzing into a human.
Besides this not much more is known about the selection process, although it seems rather apparant that she does not really have much of a checklist.
It really is a rather flawed system, what Gaia is in essence doing when there is a problem that requires a Bee is to turn humans into Bees and leave them to their own devices and hope that they will meander to whatever the problem is and MAYBE fix it.
When it comes to physical conditions and limitationsā¦ According to devs on the old forums, the Buzzing can fix most things but not everything. There are things they cannot fix but exactly what those things are the devs have not mentioned much. With one exception. Going to have to dig through the forums later so I can actually link to them for ease of verificationā¦ but anyway!
The Buzzing cannot fix your brain completely. It is apparently an extremely complex thing and the Buzzing cannot completely fix that, which touches upon mental conditions - These will not be fixed or prevented. A psychopath will remain a psychopath, if youāre crazy you wonāt magically get better, nor will the Buzzing be able to prevent you from going insane if such a thing were to happen apparently.
Which does seem a bit odd when you think about it since a lot of mental illnesses/issues are actually due to chemical imbalances in your brain so youād think the Buzzing would be able to regulate that as they pretty much keep your body in tip-top shape otherwise and keep impurities/imbalances/diseases/toxins from affecting your body.
But maybe itās just one of those things they canāt fix. At least they scarred our minds to prevent outright Filth mutations soā¦ yay?
As for physical changes such as muscles and whatnotā¦ that kind of depends on how you make your character.
Canonically Bees become faster and seemingly stronger than a normal human, stamina and endurance seem to be increased as well. Muscle mass does not seem to increase though. (Up to RPers reallly) The main powers of Bees are pretty much the ability to be reconstructed by the Buzzing to prevent death and the ability to spew out Anima both in itās raw form and in specific forms (Maaaaagic!).
Bees are still as squishy as a normal human in most cases thoughā¦ a hard enough conk on the head or neck and itās lights out! They are injured just as easily by weapons and certain environmental effects (Extreme heat and cold) as a human as well with one important caveat. Bees can pretty much reach terminal velocity while falling and when they impact the ground the ground is what will be damaged, not the Bees. (Yes, no fall damage is actually canonā¦) We even crack the ground in the Facility dungeon due to falling onto itā¦ 
Humans have free will, Gaia and the Buzzing do not change this. They do not force us Bees to do anything against our will, nor do they outright tell us what we are supposed to do.
So no, Bees would be very much a threat to their peers and those they are set to protect by whoever they happen to be alligned with if a Bee were so inclined. Bees are not made to protect humans. They are made to protect Gaia, the Engines and the planet itself you could say. Nothing more, nothing less. The only reason we actually āfixā anything in the canon story is because the human factions that picked us up actually send us towards the problems. And sometimes we just fix things through a flukeā¦ And canonically plenty of the Bees actually make things worseā¦
Gaia and factions choosing you
Thatās really down to the RPer really.
Granted, when the recruiters show up at your door they talk (Wellā¦ two of themā¦) to you as if you are not affiliated with them so at the very least one could say that canonically you werenāt outright affiliated with the Illuminati or the Templars at least. But then again, perhaps you were, just not directly, or maybe you wereā¦ all up to you.
Besides that we have no knowledge of the characterās life prior to the recruitment pitch so for all we know they might have been involved in the Secret World already.
Gaia does not seem to choose someone in advance per ce. Some people can hear the Buzzing and may learn things sureā¦ or end up insane or wrongly locked up in asylumsā¦ Some people might have powers already and learn of Gaia through connections they might make. But as far as we can tell Gaia only makes Bees when they are needed and itās entirely random who becomes one. Thereās no waiting line for becoming a Bee. (Now servicing #42!)
That is not to say you canāt be doing Gaiaās work though or at least what you believe it to beā¦ Zuberi comes to mind.
In my opinion the mass surge of Bees worldwide was a āOh ā ā ā ā ! BOMB! LAUNCH TEAM BEE! LAUNCH LAUNCH LAUNCH!ā kind of thing.
You were left alone for some time before the factions came knocking so while they do seem to have a method of finding you, they donāt seem to have had knowledge of potential canditates really. Itās pretty much a scramble to grab as many Bees as they can before someone else snags them.
Not to mentional a race to dispose of any Bees that wonāt join up with themā¦
Your cabal
Iād say itās probably a bit more relevant now with the multifaction cabals. Especially for the cabals that have mainly been running as corporations, mercenary squads and similar entities. It allows for them to actually be proper groups rather than just a faction aligned/backed entity, which is a good thing really. More diversity that way.
Timeline wise cabals can be troublesome but with the timey-wimeyness of Agartha most RPers seem to be able to handle the discrepancies decently most of the time. Itās not like we can demand RPers and groups to condense years worth of plots into a few months really. So timey-wimeyness it is!
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