Question one: your apartment should be the first thing listed on your Teleport menu if you have All or Other selected.

Question two: Right click a song in your playlist and select Play All, Shuffle or Repeat there. I don't know why the buttons at the bottom of the playlist don't work.

Question three: Yeah, some of it is confusing.

Question four: Allowing others to visit your apartment isn't a problem. No visitors can remove anything from your personal housing. It's only an issue if you have a house and give someone else tenant access with the ability to place/remove furnishings. In that instance, the tenant could take some you purchased.

Mog station items can also be lost if you own a house that gets demolished and don't claim them from the Resident Caretaker within 35 days. Not a problem with apartments since apartments have no auto-demo.