It sounds like you're asking two questions here: first, what items can the PCs pick up from enemies, and second, what properties do those items retain from enemies' stat blocks?
1) What items can PCs pick up?
In theory, everything the enemy is listed as carrying. In reality, as Lucas says in his answer, picking up everything quickly becomes hugely impractical. The players end up with large amounts of useless or near-useless items which they must haul around, and whose sale value is negligible.
As an example, opening to a random MM3 page with a range of heroic-tier cultists, I see "club" listed for one cultist (which would be a simple, nonmagical club of the kind you can buy as starting equipment); and "plate armor, fullblade" for a different cultist (again, non-magical). Those nonmagical clubs and swords are of no real use to the levels six and seven PCs that will be fighting them, since those PCs almost certainly have at least +1 items at this point.
2) What properties do enemies' items retain when used by PCs?
If the item is not specifically listed as having the ability, then the item does not have it. Just as a PC can use powers to temporarily set their weapons on fire, or make their armor more resistant to damage, enemies use similar innate powers on their items to gain the extra abilities seen in their stat blocks.
Going back to my MM3 cultists, the guy with the fullblade has a power that lets him apply ongoing 5 fire damage with a successful opportunity attack. But his weapon is listed as just a plain, nonmagical fullblade - that fire ability is innate to the cultist, not to the blade.
If your players are dubious about this, or want the abilities the enemies used, you can fluff it in a lot of ways to make it more understandable. For your kobolds, you could say that either the fire resistance is something which the kobolds have trained for, perhaps by learning how to handle the heat through the shield; or they applied some kind of short-term coating which only lasts for one encounter. (This is related to something I touched on in my question here, and a lot of the answers to that question are useful for working with your players when this situation comes up.)
TL;DR: Yes, the players can pick up everything. No, they don't get the abilities the enemies were applying to those items.