Does Skyrim follow the 'must be owned' rule or can you set up shop in certain places? There's a neat cave I wanted to use but is it safe to leave my gear there?

There has never been a "must be owned" rule. The only rule for item storage safety was that the container wasn't a respawning one.

The only ones I know for sure are the houses you can buy and the room you're given for joining the Winterhold College.

To know if a container is safe, loot it, wait 3 or 4 days in-game, and then see if the container has more loot in it. If yes, it's not safe. If no, it's safe.

Once the construction kit comes out, you can mod your cave to be a safehouse.

why not just drop your stuff inside your house? theres plenty of them u can buy, i don't get it.

This is not a good idea in Skyrim.

NPCs can, and will, take things you drop on the ground. They even have events that can play out if you do. Like dropping some armor on the ground, a guard picks it up, comes up to you and gives it back telling you not to litter. Drop a sword and a kid will pick it up and ask if you wanted it back. Drop a jewel in a bar and watch the patrons fight over who keeps it.

Edit: Nevermind you said INSIDE your house. For some reason I read it as "outside." O.o

He is probably confirming before dropping off named items that can only be obtained once.

Dragon skeletons can work, but they have a bug where the spawn all over the place sometimes, not where you left them last.
I used a barrel in the companions' hall for a day or two as I saved for my first house. Now I have my furniture filled with books and ore.

I usually drop it inside of barrels around white run, works for me. None of NPC's will take it out.

oh I know... Can't wait
My character isn't one for living in cities...