I'll leave #1 to others, as I haven't had to deal with repairs. At all.  I usually have ships drop out and upgrade to restore their HP.

2) Yes, the higher your diplomacy vs the other side's diplomacy, the better the deals you can make.  The positive bonus to relations is fairly minor in comparison.  Indeed, I sometimes can garner such a staggering diplomatic bonus that I can flat-out trade an inferior tech for a superior tech, and hit the credit cap on tech for cash trades.  This is why so many people swear by the Diplomatic Translators- it's a big Diplomacy bonus that you want to keep to yourself, if at all possible, and obtain through any means if someone manages to beat you to them.

How about if I rephrase the question as "Does a ship that isn't orbiting a colony repair more quickly if it does not use any of its movement?"

My instincts say the answer should be "yes," but I've never methodically tried to get an answer 'cause asking questions is much easier 

Okay.

I hate repair.

Seriously.

I hate it more than I hate military rating.

...okay, maybe not that much.

I just had a ship sit here for 12 turns before it repaired one point of hp, and I have no idea why.

-

It would seem that repair has a chance modifier in addition to a value modifier.

Being in orbit appears to set the chance modifier to 1 (in other words, a ship will always repair when in orbit, whereas out in space, more often than not it won't on any given turn).

It also seems to double the value modifier.  In orbit I -generally- see a repair of 2 hp per turn, regardless of ship hp, whereas in space it's normally 1 when it happens.  I think there was a turn where I saw 3 hp repaired in space, but I may be mistaken.

Which brings me to my other conclusion-that repair is not percentage based.  You can stop putting a gazillion hp modules on your ships now.  (Those who upgrade-repair don't even need to, really.)

So, given the above, it is extremely difficult to ascertain whether or not using up the moves of a ship has any effect whatsoever on its ability to repair.  However, in testing with moving around my homeworld and then dropping into orbit each turn, I'm going to assume that's a no.

Additionally, there does not appear to be a difference whether I'm repairing in my territory or an enemy's (Yor, if anyone's curious).  But, again, since it's a chance to repair in the first place, that's really hard to test.

For what it's worth, the chance modifier seems to be in the realm of 20%.  It'd be real nice to have official word on that, though, and that's just my best guess, as I'm already annoyed with it.

Nor can I, but I don't see any other explanation for my ship sitting in space for twelve turns without repairing a single hitpoint.

At the very least, it gets around the realism issue of the repair rate optimally being higher when in orbit than in space.

And in any case, the repair "rate" is 100% when in orbit, which is the important part.