Can I just pick any shelf in any home and use it to store things that I don't want to carry? I don't see how this could be a problem, as Fallout 2 does not use Leveled Lists to spawn random items, but I thought I'd check with you guys.

Its perfectly safe.

Should be ok. In-building containers rarely have any sort of respawn script attached, especially if it's a deserted building. (If it's not deserted, it may have ownership flags to someone.) The car is probably better storage, though.

I'd also recommend the car boot, even if you havent purchased the car yet.