Are there by chance any mods that make the npc behave more like in Oblivion and Skyrim, such as visiting shops, sleeping, eating etc? I found a mod called "Living cities of Vvardenfell" that gives a schedule to most of them, but they don't actually "walk" when changing locations, but just teleport. Are there any mods that have animation of them walking through doors etc?

No, as this is not possible in Morrowind engine AFAIK.

Living Cities is your only choice. Morrowind's AI and animation packages are pretty limited.

There is http://mw.modhistory.com/download-13-6153 which allows NPCs to follow the PC through doors, but as with most MWSE mods, it might be a little unstable.

This looks interesting. Have you tried that mod?

A long time ago, I don't remember the results. But I've recently lost faith in MWSE.

Pursuit Enhanced works as advertized. Enimies do follow you and it adds a whole new dimension for stealth charachters. You no longer are safe from enimies behind doors. But, the only annoyance is too frequent crashes, especially during battles and while loading new games. I've come to learn that MWSE or MWE mods which constantly keeps track of references are too unstable. Whatever MWSE is capable of, its still a sort of a hack to be a part of the game to work smoothly.

I wonder. Would it be possible to make NPC's follow you through the building, then return to the starting position? That would be a nice touch, also preventing the player from stealing stuff

I've put my hopes in the possibilities OpenMW will bring for implementations like this.