Re desoldering,
Quick jobs
4-6 leads- chemtronics soder-wick (kept sealed) and twiddle of the lead after. My most used technique.
2 leads and I have a spare, cut it out, desolder one lead at a time, either solder wick or tooth pick to clean up the hole after.
Big jobs eg 8+ leads
Get out the desoldering gear,
An el-cheapo zd 987 works quite well BUT as said before here, keep your finger on the trigger to suck the solder through, and keep cleaning the chamber and nozzle/channel.
I have a Metcal desoldering gun, it works very well but you have to fire up an air compressor. It can suck solder out of a via in a heavy solid ground plain. I use it rarely.
Solder suckers - not my favorite, haven't got one of those better ones though.
I have tried the hollow needles, often you don't have the room to deploy them.
I have used a hot air gun with mylar tape on a few smd devices - went ok, only a couple of wayward nearby components.
Re broken studs etc, I am glad to see that it is not just me who has repair jobs like that! My daughter's front wheel bearing replacement was a saga of epic proportions. I still have nightmares about that car - it's the anti-Christ! It won't die!