I am not a lawyer, but I think that as long as you are not hosting the content, then there is no problem. e.g. Adobe can hardly be held responsible for what people make in photoshop, be it copyright infringement or even a criminal act (although interestingly Photoshop supposedly includes code that attempts to catch currency forgers)
As an example, there are plenty of sports games without licenses that let you modify the names of teams and players. I believe this is perfectly legal, even though it's obvious the feature is there to let the player base use licensed names. Such games often allow methods for players to save and share their modified team rosters, this is perfectly legal. However, if the method of sharing the rosters involved the developers website/servers in any way, then most likely the developer would get sued. i.e. Let your users save the data as a file and share it through their own website and your fine, provide the service that lets the users share the data and you're in trouble.
Here is my opinion on your specific questions:
-I guess allowing them to change 3D models and changing textures is not a good idea. AFAIK, Im responsible for whatever content they create. Right? You are not responsible
-What about smaller things like names, color templates, etc. You are not responsible
-What about a map editor? Edit the heightmap/tiles, place the built-in objects etc. You would only be responsible if you run some infrastructure that allows players to share the maps
-Is it any different when the game is single-player/online multiplayer? If you are hosting multiplayer servers then you would be responsible for making sure user created content is appropriate. If you're not hosting the user created content then you must still make an effort to warn players that they may (i.e. will definitely) be exposed to obscene content in multiplayer play.
-What if they create a mod for your game? Are you still responsible for its content? You are not responsible unless you host the mods on your website or something
-What if I dont explicitly allow them to change things like textures, sounds etc, but they can change the files in the game`s data folder? You are not responsible
-What if I do protect the files with checksums or something, but someone hacks it? You are not responsible, but if you ship something inappropriate which can be accessed through hacking then you are responsible (e.g. Hot Coffee, or that guy that padded a console DVD with South Park episodes instead of with zeroes, it was inaccessible except by hacking, but they still had to recall all the DVDs)
Again I am not a lawyer, I may be wrong about any of this.
Edit: I suppose the City of Heroes vs Marvel case is relevant: http://en.wikipedia.org/wiki/City_of_Heroes
Edited by C0lumbo, 23 February 2014 - 10:37 AM.