Where can I safely store my stuff?
If I store things in a chest in a room I have rented, will it safely stay there like it would in a house I own or will it vanish after a few days?
Items in any chest that doesn't belong to you can vanish.
Most containers in the game are subject to reset, that is, they clear out all the contents after a certain period of time and are reset to their initial state. Most containers reset after about a week of game time, however, examining the contents of the container causes the countdown to start over. A container could be made to never reset simply by opening it once every 5-6 days or so. Any item a player has left in a container that has reset is lost. Wiki
(Just on a side note, I read a few times that chests had reseted earlier than the 7 days the wiki states as the interval.)