Do you know any vanilla doors that after you open them stay open for a few minutes then automatically close? I want to look at them in the cs and see how they do it.

I've been thinking about this as well, it's quite pointless to have doors when the NPCs just open them right away.

I don't believe there are any vanilla doors that do this but there are some house mods that have auto-closing doors.

Do you know the name of the mod?

http://skyrim.nexusmods.com/mods/18575 is one...if I remember correctly.

http://skyrim.nexusmods.com/mods/14152 has one in the Garden room. It uses a triggerzone so when you leave the vicinity, it auto-closes. It will also auto-open for the player and NPC's.
The gate in the Arena will only open for the player. It also closes when you leave the vicinity of the gate.