How long does it take for stolen goods to stop being considered hot? I'm talking about that little red icon near the items. Normally, I would just sell the item to a fence but this is a key I'm talking about. I've waited and I've slept but the icon has yet to leave.
They never go away. The only way to get rid of them is to sell them to a fence and buy it back (although it costs a lot). Items never cease to be "hot". Although it doesn't really matter unless you are trying to sell them to honest merchants. You can parade in a fully stolen outfit in front of guards or the owner of those items and they won't say anything.
Guards wont approach you if you only have stolen goods. They only aproach you if you have a bounty. Check on the statistics page. The only way to get rid of it is to do what the guards say, befriend one so he pays the fine for you, or get armand or skrivva from the thieves guild to get rid of it for you.
If you check your stats, I think you'll see that you have a bounty on your head. You'll get a bounty if you murder someone in cold blood, or if someone catches you stealing. You can pay your bounty to the guards or go to jail. You can also make the thieves guild doyens take care of it for you, for half the price.
I used to play Oblivion, then I took an arrow in the knee.
You know the character page? It's the first page in your journal, where you see your name and class and level and etc... The first tab of the journal is divided into four (or five) smaller tabs. The last one is the statistics page. It keeps track of random statistics like highest bounty, items stolen, jokes told, etc. Scroll down on that and it will show you your bounty somewhere.
For 0 gold items, chances are that they will be considered stolen forever, unless it is something you can enchant. If you try and "give" those to a fence, you won't be seeing that item anymore as they won't sell it back.
If it was me i would place the key in my house or a bin of some sort, then go pay my fine or get arrested. After they let me out, I would hurry back to get the stolen item.