I haven't looked at the kart yet, but I think I know why you had to add a solidify modifier to fix invisible faces.
Every face has a forwards and backwards direction. It's called the face's "normal". In order save on computing power, game engines will generally only render a face if it's facing the camera. So if the faces in your kart were oriented the wrong way, they wouldn't be rendered.
To check if this is the case, you can use the "Show Normals" option in edit mode. It's found on the properties panel, which you can open by pressing "N".
To fix it (in Blender 2.71), you can select the faces with incorrect normals, go to the shading tab in the left panel, and click "Flip Direction". You can also try "Recalculate" (ctrl-N), which will make Blender try to fix the normals for the current selection.
EDIT: I just looked at the pictures, and I think that the kart's really not bad at all. Textures are decent, the shape is original, and the poly-count doesn't seem that high. It's definitely a much better first model that what I started with.
The one issue that I did notice, however, is that the stripe on the side seems to have been handpainted. I'd recommend redoing it in a vector graphics program, or at least using a straight line tool in a raster editor.
Altogether though, I think this is a pretty solid kart and a great first model.