Is there a way in Blender to check a mesh for missing or duplicate faces, mesh abnormalities, or other issues that might cause problems when it is exported and used as data for a real-time application such as a game?
There is an addon called Mesh Lint that allows you to check a mesh for several of the problems that might cause issues in games. You can find more information about it here: http://wiki.blender.org/index.php/Extensions:2.6/Py/Scripts/Modeling/MeshLint
It should still work with current (2.7x) versions of Blender, although it is not included, so you will need to download it from the link above and install it manually (user File > User Preferences... > Addons > Install from File...)
There is a video about how to use it here: http://cgcookie.com/blender/2013/07/23/addon-mesh-lint/
There's an old way. In edit mode, select nothing with A and then hit ctrl alt shift M to select non manifold verts or edges. These are edges that do not have exactly two faces connected to them. If nothing gets selected, your mesh is pretty solid.
Try it with vertex select or edge select. Both modes provide insight.
In addition to the mesh lint answer I provided above, Blender also has a couple of functions that can help solve common problems. I recently switched to Blender for game work and these are the functions I have found to be invaluable for preparing mesh for export.
Remove duplicates / merge vertices: To remove duplicate faces, vertices or edges in a mesh, enter Edit mode, select everything and press W to bring up the specials menu. From this menu select Remove Doubles (or press 4) and any duplicate elements will be removed.
Note: this function will also merge vertices that are in the same spot or very close together; the threshold can be configured by changing the Merge distance setting located in the Last Operator panel in the Toolshelf (or press F6 after operation executing).
Select Linked: You can select all parts of a mesh that are linked together by selecting one part and pressing Ctrl+L. If you then invert the selection with Ctrl+i, you can find any unwanted loose polygons.
Select Loose Geometry: This will select single mesh elements that are not connected to others, such as floating vertices, edges and faces. Note that you need to be in vertex edit to select vertices, edge to select edges and face to select floating faces, and it won't work if there is more than one element connected together, which is why I prefer Select Linked.
Apply Transform: You can apply (or freeze) any transform on a mesh by pressing Ctrl+A and selecting the desired option. This can help to fix problems with scale / rotation after export.
Move Pivot: You can move the object pivot with Shift+Ctrl+Alt+C. The pivot can be moved to the 3D Cursor, which can be moved exactly with Shift+S or by entering a position in the Properties panel (N) in the 3D view. This can fix problems with a model being in an unexpected position or rotation after exporting.
Yo Omegacron! Blender also has a cleanup function. It has some functions that are similar to the CleanUp in Maya like 'generate dissolve' (deletes zero length faces and edges, also double faces) You can find it if you go to Edit Mode and to Mesh -> CleanUp in the 3D View Header. Make sure you selected the whole mesh before you clean up, generate dissolve, delete loose and fill holes should make your mesh game-ready :)