Cleaning up the leftover geometry depends on whether or not the unwanted 'bits' are whole faces, in which case you can zoom in and erase away. Otherwise you could turn on hidden geometry and add lines in strategic places to keep the parts of faces you need, and then delete the remainder. Someone with more experience than me could direct you better here I'm sure.
Regarding the corner you mentioned... hard to tell from the pic what would be causing the texture issue. If you haven't already done so, you might make sure all of your faces are faced correctly (front face facing out). I've read that reversed faces can cause rendering issues, so maybe your game is reacting the same way.
Good luck!