Lark, there can be many steps involved. I'll try to help with some. I've had hit or miss success with exporting as .model. Some are just fine and only need recalculate normals & tangents in GMT, others create the errors you're getting (and I've done the same things to both models, setup same structure, etc., could be me though

).
Some help...first, select the entire mesh in edit mode and do Mesh -> Vertices -> Remove Doubles. Blender will flash a message at top center of the screen saying how many verts have been removed. You have to do that for every .model you import. If you export .obj from Blender and the faces disappear, easy fix, the normals have been flipped. Go into edit mode, select all the problem faces then do Mesh -> Normals -> Flip Normals.
I just tested importing the dungeon_wall_alcove and exporting as .model into GMT and it's fine. I have no problems exporting .model as long as no changes are made. Is it possible you have leftover objects in the scene? Open Blender, hit delete and get rid of the cube, maximize the window, then press Ctrl-U and select Save Startup File? Now you always have a clean scene when you start, try the alcove again.
I haven't tried to texture an imported .model yet so I'm not sure what's happening there.
It can be tricky to align the Blender and Grimrock worlds. To get your models positioned correctly, you have to have the game_matrix & RootNode in Blender setup correctly, you also have to assign textures to their respective meshes. Remember that Blender has the vertical axis as z whereas it's y in Grimrock. If everything is setup correctly, you can export as .obj with only needing to recalc normals & tangents in GMT.