Do the dying soldiers live if I end the mission before stabilizing them

Cars indeed have a damage threshold, 2 of them, even!

First threshold prevents it from catching fire or exploding. (Pistol hitting a car for example) Then once that one is passed the car catches fire. The second one is where it explodes straight away.

Things to remember when using cars for cover or shooting at them:

• Cars caught fire during your turn explode on your next turn.
• Cars caught fire during an enemy turn explode on their next turn.
• Cover remains after a car blew up!
• You can use grenades and rockets to target and blow up cars before using their cover.
• Kills done by cars are like grenades and rockets they yield no materials only a corpse (fragments)

From what I read, others report that grenades kill cars instantly, where in my games they never do that. It seems higher difficulty is higher threshold. I play impossible so as such I will not note my damage expectations and experiences.

Your third example is a bug, I believe. I've run into it, it's super annoying (and occasionally devastating).

Unfortunately, I have never seen anything from the UI/options that would help in either situation #1 or #2. About the only thing you can do in these situations is to move to the closest possible cover within the blue, and then if you can't take the shot you want, either go to overwatch or possibly re-position as part of the dash with the goal of moving into a more advantageous position on the following turn. The cover thing is especially vexing, I feel your pain there. The game is forcing you to eyeball it; unfortunately, cover is at high enough level of abstraction such that eyeballing it is not always possible.

Note that Assaults, once leveled, are really good to use in these situations. You can move to wherever you think is best in the blue radius and then if you're unsatisfied with the results you can pop Run n Gun and advance further and then shoot (and you can even do Rapid Fire after doing a run and gun, it's quite a potent combo).