Doing Mesh vs Mesh collision detection in managed code is quite an overkill for any bigger project. I assume you have your "character" which should move around, and if so, it is probably even animated. That means you have to do collision detection almost frame-by-frame, and that, IMHO, WILL kill FPS.
Now lets look at things which were already done. For example, games written in C++. They do not detect collision between visible meshes, but ussualy use some kind of optimisation. For example, Unreal2 engine games uses cylinder around players (HalfLife2 engine - infamous simplified colision mesh). Players ussualy move on three surface types: terrain, BSP geometry, and meshes. For terrain, players are "glued" to it when not jumping. BSP is basic sculpting tool (which can be used to make basic shapes of buildings, etc), and contains small amount of polygons (in other engines it could be mesh). And finally, there are StaticMeshes, which are used for decoration. Furthermore, artists ussually make simplified collision mesh almost for every static mesh, which most of the time happens to be box. And this is for C++ game.
I am not saying that this is the way it should be done

. But for simple game or project cylinder (or box, or few boxes) around model should be fine. I am sure that Mesh Intersect function WILL be included in later XNA version. However, in such case simplified collision would help to achieve even higher FPS

Now, how to detect that simplified mesh... I looked around a little and so far i found
this solution, to collect information about mesh part by drawing vertices as points while writing info about that somewhere. Also, i don't know if you already saw this, but someone is writing
XNA tutorial exactly for doing that. I am going to look at his custom processing tool when i have more time.
For doors, there is one more way to make them work - place object "Trigger" there, which will make them open when character enters specified radius (or cylinder), and close when he exits it.
Well, sorry for not posting any real sollution, but maybe some info will be usefull...