I've seen a couple of different weapon chests. Two from the original game, one that looks like a car trunk, etc. Do any of these containers have an higher chance of containing higher level items? Do their contents scale with your level? Do all players see the same loot?
Yes, different items have different loot. Most have "minor loot" (ammo, cash or health) but certain containers always have guns. Containers that always have guns also tend to have better weapons on average.
Chest contents generally don't scale with your level (though in Borderlands 2 they did in playthrough 2.5), however the Golden Chest items do scale with your level
Stuff that has guns
These chests always have guns and the items they drop are noticeably better than the typical gun drops from baddies or "non-gun" containers
Stuff that has minor loot
I'll try and get pics on these while I'm playing
All players in coop see the same loot except when it's mission loot (you get slightly different reward items).
With regards to containers that can only hold certain items:
You'll also see a bunch of exploding items (gas tanks, gas canisters) which have a green light like unopened chests. If you blow these up, they drop ammo and money, and (possibly, I can't remember) a common item.