As @SIsilicon28 said, you need to turn back face culling on. In blender, press N, and a menu should come up on the right side of the screen. Somewhere down the menu, you should see back face culling. I always have it on by default; if you want to save the start-up file, that's CTRL-U.
I think the extrude problem is simply because you extruded the wrong direction. You can see problems like this if you have back face culling on, but you can fix it:
Select all the faces that need to be flipped
Press W
* Flip normals
Flip every normal that needs it. You will know which normals need to be flipped when back face culling is on.