For the first part of your post, who is the one setting up the circumstance, the player or the GM? My answer will be pretty similar in either case, have a discussion with the other party, but it'll be slightly different depending on who is initiating the conversation. If it's the player, then prepare to negotiate some give with the GM, or undergo some task or other in game they may want you to perform for the sake of catching the deity's favor. A boon is a power increase above and beyond what other players will get, so make that understanding known to the GM and explain that you are prepared to help them make the story more interesting for everyone at table in exchange. (I mean moreso than people should already be doing, since Pathfinder is a cooperative and collaborative game.)
As for your enumerated list,
1. I'd primarily look at how long-term the effects will be on the game. The boons that grant an auto success at certain rolls are generally pretty great, because they're one and done, and also mean the party can test themselves and push bounds on checks they would normally have much less chance of succeeding at. If the boon is handed out for something difficult that party member did then it also feels like a reference back to their hard work, which is good. Honestly even some longer-lasting boons can be minimally impactful to the game, like Shizuru's boon of granting life-saving armor, depending on a game's level of deadliness.
Then there are some divine boons, like Brigh's last boon, that can very much shift the power of a game. I picked Brigh's because her major boon grants the party an ally construct of one level lower, meaning that the party both gets another actions' worth of stuff to do, which is big, but also that those actions are close in significance to the party's own, which is not to be underestimated at all.
2. I think that's a bit tricky to answer because boons aren't built equally. They were built for general thematic consistency, instead, so saying something like "wait until level 5 before granting a moderate boon" doesn't make sense. Different moderate boons are differently significant. Here are a couple very general rules I'd follow if I were handing out boons, though.
I think the reroll boons are safest. Pass those out at any level, especially if you know the party is going into an encounter or situation they have relatively low odds of succeeding at. By and large boons that grant extra flexibility to attacks should prooooooooooobably start showing up around the same time that elemental runes become available, and the same goes for boons that grant extra defenses. Boons that help with skills are likely the trickiest because their effects are all over the place, from granting bonuses and feats to tipping the scale of success up by one. I'd actually lean toward handing those out at lower levels because anyone who is using said skill a lot probably wanted to make a character invested in that skill, and benefits to skill actions are easier to notice at lower levels. I would probably not start bringing major boons into a game before level 11 myself. (Again though, it's possible a character could make some great sacrifice or accomplish some great feat in service of a deity, so the levels can and probably should be fudged in service of the story.)
Honestly I think the biggest thing to keep in mind is to be consistent. If a player is getting boons, try to find something equally cool to give the other players. Boons are a power increase, and while it's fun to hand them out as rewards it's also possible the other players at table will see it more as a punishment for not playing to the GM's expectations than they will see it a reward for someone who managed to pull off something cool.
3. I think that if a player wants a specific boon then they should absolutely talk it over with their GM. I as a GM personally love when my players have goals to work toward. It makes my workload thinking up plot points easier, and makes the adventure more personally interesting to them.
That being said, no I don't think a player should expect to get a boon. They're a power increase over other players, like I said, and I think any GM is within their rights to refuse that power increase if they think it'll cause problems. I also think that, yeah, boons should be handed out in play for the most part. The whole point of a boon is that it's a cool way for the gods to show favor, or I guess disfavor, in play. If the boon is happening off-screen then its specialness and narrative impact get diminished and it becomes just another mechanic to be manipulated. That feels like it goes against the spirit of what boons are meant to do, to me.
