In lieu of a house, I'm considering clearing a dungeon and storing stuff in its chest, but I don't know if the enemies will come back. After killing everything in a dungeon, will more enemies respawn eventually?
Within a couple of days, there is no respawn. Respawn now takes 30 ingame days, which is longer than in previous ES games.
You even get a "cleared" marker on the map when you eradicated an area of all its inhabitants for the duration.
However, when you have not fully cleared sites, enemies may respawn. I experienced this myself at some fort which name I can't remember right now. I killed 3 of the guys outside, went into a tower inside the fort, came out again and they respawned.
By the way, storing stuff inside containers which do not explicitly belong to you (e.g. stand in your house) doesn't seem to be safe. At least one of the loading screens explicitly says that storing anything in "wild" containers does not guarantee for it to be in there when you look for it the next time.
One way to find a dungeon which has respawned enemies inside is to accept one of the various kill quests that you can find (from innkeepers, the Dark Brotherhood, etc.), which will lead you to you to these dungeons. These procedurally generated kill-quests will send you to dungeons where the difficulty inside matches your level, which means that taking one of these at higher levels will have a higher likelyhood of taking you to a more difficult respawn-dungeon.
I have been playing the game for a while and though some dungeons don't respawn (typically dungeons tied to major plot lines) most do and in very short order (maybe a week tops). I've cleared one giant camp probably 5 times. I could not give you an exact figure but the rate at which they respawn seems to vary somewhat from place to place. The items in both chests and lying around on shelves do respawn and are leveled, and any items you leave in a dungeon container will be gone when it respawns (theoretically I haven't bothered trying but there is a load screen message which says as much and it was the case in oblivion)
I've gone back to dungeons I couldn't finish earlier because they were too hard, and all the enemies and urns/shelves/containers respawned. The keys I had used to unlock the doors didn't respawn and the doors were unlocked.... but all the equipment I had left on the ground in the dungeon was still there.