The factions are mostly what you make of them yourself. The vast majority of the game is the same for all the factions - there are some where the story set up for missions varies depending on your faction, otherwise there’s only 3 missions that are unique to each faction.
The Templar connection to the church is tenuous at best - the Illuminati are just as likely to be involved in the hierarchy of any religion, and the Dragon will have strategically placed agents too. There’s rigidity and flexibility in all of the factions, it mostly depends on how you want to approach it.
The Templar mission that Sawo mentions (Virgula Divinia) is cool, but there are other little bits which the other factions get that Templars miss out on. If you like the smurfs or the lizards, pick them over the red-shirts
Starting weapons are dictated by the starting class you choose. Each class has a designated primary and secondary weapons, and you can then unlock additional weapons using in game currency as you progress. There are only a limited number of classes, but your class is not restricted or limited by your faction choice.
There are 9 starting classes (basically one per primary weapon), but if you want a specific combo then the first extra weapon unlock isn’t expensive to buy. Warlock (Blood Magic primary and Assault Rifle secondary,) or Ravager (Fist Weapon and Blood Magic) would seem to be an easy option, and would just mean you had to buy chaos.
It’s worth checking out https://www.tswdb.com/miscellaneous/the-ultimate-beginner-guide/ for a really good starting resource!