Q1: Yes, its every time you FTL jump (you can hire a mercenary to delay them for a bit, though I've never made use of that)
Q2: As said, Its only in Combat with enemy crew members (hand to hand, or using their teeny little laser pistols). It doesn't affect ship damage if they are manning the weapons.
Q3: The crew member who finishes the repair/combat action is the one who gets the experience.
So for instance, if 3 people are fixing the room, the person who gets the last bit of a power bar fixed gets the experience, regardless of how much of a contribution they all made (obviously if 3 people are fixing it at the same time they will "finish" it at the same time, so I don't know how you would tell who gets the experience, just know that only one of them will. If you wanted to make sure, you could move 2 away leaving just one guy left and he would get the experience for sure, even if he only came in at the last minute)
Note that you get 1 repair experience for EACH power bar of a system that is fixed, so if your weapons had 3 damaged bars, then your crew would get 3 repair xp total (but it would be divided between the repairers)
I think that Breaches and fire don't count (sadly), though a better repair skill can repair a breach faster (not sure about fire, I think so)
And someone said somewhere on the forum that shields are a bit special, and only counts for 1 xp per 2 bars (given you can only power 2 bars at a time) but I'm not certain on that.
For combat, the person who deals the killing blow gets the xp, and no one else. You get 1xp for killing 1 guy, however you ALSO get 1 xp for each power bar of an enemy system you destroy, so boarding parties can level up pretty quick that way. Provided you don't blow up the ship they are on...
And yes, throwing more people at a problem will combine their repair skills