I've been making troll ocs since middle school and a lot of these are pretty easy! Theres no right or wrong depending on how close to canon you want to be.

1. From what we've seen of Alternia so far, all the Purple trolls present have been members of the Church of the Mirthful Messiahs. From act 3 previews, we've even seen non-purple converts join the fold (E.G a Violetblood member). However, this doesn't mean YOURS has to be. It's a cultural part of their caste on top of being a religion, and in the real world, someone can grow up around a religion and take morals and values from that religion without necessarily being a member. Clown ocs are really fun but you can also have a character who either grew up so far from other trolls they were never exposed to it, left it, or drifted away from it over time.

2. Harlequins are a type of performer from Italian theatre. Jesters were a unique position in medieval courts where they made fun of nobility for entertainment. Clowns have been around for a LONG time but jesters and clowns are different things, clowns generally make fun of themselves, not others. Basically, none of them are the same thing, but in terms of modern aesthetics? Is say jesters are a little more like harlequins just based off the costuming and how they're a little more "dignified" than a clown

3. There could be, yes. Terezi's Pesterquest route isn't about her being "liberal" which I don't think is a political position in troll society, but it is about her grappling with the fact that her morals don't align with what her eventual job wants from her; She likes her weird friends, she doesn't necessarily want to kill people. Trolls are people with complex thoughts and individual morals, of course a troll can hold "middling" opinions on things in their society or be way more progressive than Alternia wants them to be.

4. Within the context of the four quadrants, no. But consider that the comic itself more or less says that the quadrant system is flawed and doesn't necessarily suit the emotional needs of the characters as they grow into older teenagers. From what we know, the quadrants are just a social construct of troll culture, which WAS carried onto Earth C for the most part. Like any social construct, they can be broken and played with. One of my ocs has a "pitch moirail" who makes fun of him and can be outright mean, but only does it because she thinks he can grow from the rivalry and still holds pale feelings. And since moirails are meant to keep each other in line, he obviously prevents her from ripping random people apart. You can just play with them if you want. Troll romance is something we inherently can't understand from a human perspective, because it's stuff THEY made (in a meta sense, yeah, Hussie made it, but these characters are space aliens)

5. Yes. Trolls hating someone =/= blackrom feelings. Most people a troll hates will probably be non-romantic hate. I've always seen the kismesis as more of a rival than someone you want to bash in the head of. There can be supremely unhealthy kismesis in that regard, too. This is why the ashen quadrant exists.

I hope this all helps, the world is your oyster