Why aren’t the soldiers getting promoted

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.

If you have soldiers that show up as "Wounded" but lack a number of days until recovery, that soldier is, unfortunately, bugged.

A similar thing used to happen with SHIVs and was fixed in the 10/11 patch, so until Firaxis pass along another patch, you don't have any real options.