There's nothing you, as a player, can do in-game to make any door auto-close. There are a collection of scripts that get applied to a housing door that do things like manage the conversation and interactions that can happen. If the door is not auto-closing at all, either when going in from outside or out from inside (or both), then this is likely a bug and one that's been around for a fairly long time. Assuming this is the case, it should be treated as such and reported as such (i.e. in the "Bug Reports" forum) with suffient information for us to do something about it (address, conditions, etc.).
Bear in mind that this statement applies only to the main door of the house...that being the one where you get the choice to open, unlock, knock, etc. If this is an interior door, then it might not be a "bug" but either an intentional decision or an oversight by whoever made the area. This is especially true for some of the customized houses that players have requested, in that they have not always wished for the interior doors to auto-close.
If this is a recent occurance and it applies to the main door of the house, then it's possible, that it has been force-unlocked by a GM, as the key scripts for housing doors are triggered by the door failing to open, meaning it is mechanically locked at all times, even when "unlocked" in the door's menu.
But again, there's nothing you can do in-game to make it auto-close. If you want us to look at it, we need more than what you've supplied so far.
With respect to the floor areas that are inaccessible, what Alatriel said is correct. Unfortunately, some times in some tilesets (and these are fortunately rare) are buggy and have resulted in inaccessible portions. We presently lack sufficient expertise in tilesets to correct such issues, so the only way to fix them is to replace the specific tiles with tiles that are not buggy.
