1: Yes. It's called a "Boolean Operation". There are at least three methods in Blender...Intersect, Union, and Difference.
2: Check the direction the "normals" are pointing. Looking at the Blender 2.3 guide, it says that selecting all the faces, then hitting Ctrl-N and choose "Recalc normals outside".
^_^
"We've got a blind date with destiny...and it looks like she's ordered the lobster."
