No, NPC thieves won't break into your chests, but yes, items you leave unattended in chests may expire. The only safe chests are ones in a house you own. You can buy a house in Whiterun for 5,000 gold.
When your follower "dies", they enter a pre-death state that generally allows them to come back to full health immediately after battle. However, if they continue to take more damage after entering that state, they will die permanently.
If there's a chance your follower is still alive but nowhere to be seen (for example, if they entered the pre-death state and you ran off, thinking they were dead dead), fast traveling to a location should cause the follower to appear back at your side.
If fast traveling still doesn't produce your follower, they're almost certainly fully dead. You can confirm this by trying to pick up another follower: if they agree to come with you, the other follower has ceased to be. Sorry about that.
The only thing you can do at this point—if you're not keen on loading up an earlier save or using console commands—is to find where your follower died and loot their corpse.
If you are amenable to using the console, generally you still need to find where your follower died to ressurrect them: when you do, open the console, target their corpse, and type:
resurrect 1
and press enter.
However, you can get around even this restriction by selecting the follower by using the prid command anywhere:
prid <RefId>
Where <RefId> is the ID corresponding to the follower, found by using the Creation Kit or by looking up the follower on UESP. Once the follower's selected, use the resurrect command mentioned above.
Best Answer
According to this wiki of Skyrim console commands, you can select the NPC in question (generally using your cursor) and enter the command
IsEssentialwhich will return 1.00 if they are, in fact, essential.