Strange mesh behavior -- cannot texture, and no vertices

I have two meshes in a project that has been working fine that are now exhibiting distinctly different and incorrect behavior.

The major problem is that one of the meshes does not show a texture in Object Mode or when I export an FBX. It does show the material in Edit Mode, and my UV unwrapping work fine; however, after I unwrap it and move to Object Mode, I don’t see the material on the mesyh. The texture in question is an image. Strangely enough, I can change the diffuse color and it respects that; however, it does not show the texture that is applied to that material except in Edit Mode. I also notice that when I select that object, there is a yellow dashed line box that appears, out of alignment with the mesh. I have a few similar projects and haven’t experienced any of these anomalies; I have attached a screenshot that hopefully explains better.

The lesser problem is that the mesh that makes up the tops of the walls show their material in Object Mode, but in Edit Mode, I don’t see any vertices, edges, or faces. If I AKEY to select all, it selects nothing – however, it shows me next to the Blender version number that I have several hundred of each, with 0 selected.

I’m concerned that my file has become corrupted; if that is the case, at least if I know that, I won’t feel too bad about re-creating the level. However, I don’t want to scrap it if the problem is just a lack of knowledge about what I’m seeing. Thanks for any help.