I'll start with the first problem...
has solutions only if b and c are parallel, that is, if for some constant . So, we set . We can decompose x into a piece that's parallel to a and a piece that's perpendicular to a, i.e., with and for some constant .
Since the dot product is linear, this gives . Thus, we see that u can be any vector we like as long as it's perpendicular to a, and we must have . The set of solutions, then, is
for all u orthogonal to a.