[Just picture it as a triangular pyramid with all sides equal]

I think that the first question can be answered without any calculation. (red and blue lines to show the two planes CDEF and ABEF respectively, better)

For the second one, I suggest you assign some coordinates to the tetrahedron and use the equation of the line AC with the normal of the plane ABEF.

For the last one, it can be seen that the line AC (broken line) and line EF (green line) don't meet in the 3D space.