Still kinda new to adventure mode and Im not sure how reputation and rumors fully work. If I was a follower of a giant termite diety, and went to the termite temples and asked for their troubles, would completing those quasi-quests improve my standing with all members of that order? With all followers of that diety? With everyone in general?
I think there isn't a split between religious order specific quests and general civ quests at the moment, so completing them and talking about them would improve your reputation with everyone in town and not just members of a specific group.
Could I become a heath person to that order, group, or temple?
You can only become a hearthperson to lords and ladies, a lieutenant to bandit leaders, or for dorf civilizations if you meet a captain of the guard you can become a fortress guard. For all of those you need to have some notable accomplishments, i.e killing bandits, beasts or criminals - you also need to talk about said accomplishments with that person. Even then it's not really guaranteed they'll let you in - sometimes they'll say there's "no room for you here" - means their "squad" is full - unfortunately you can't check how many people are in it to find out if it's worth even talking to that lord/lady/captain/ringleader.
Can I become the pope, per se?
At the moment you cannot become anything else than a hearthperson or a lord/lady. Occasionally there might be a glitch where you randomly get assigned to be something like a manager, broker or lord, although I'm not sure what causes it - probably something related to the previous person holding the position dying.