Well, for (a) I have no idea how to extend, I feel that there will be a continuous extension. For example I can define $f=g$ when $f$ takes values from upper boundary of the disk and upper half plane and same way for lower boundary and lower half plane. For (b) I have no idea.
(c) is true by Uryshon's lemma? As the plane is normal, and the given sets are closed and disjoint.