your first question is easy to answer though:
suppose that is a finite dimensional vector space, an operator on , an eigenvalue of and the generalized eigenspace corresponding to
suppose that is an eigenspace. then there exists a scalar such that . now if for some and
, then and, since we'll get thus which implies
so the only way that a generalized eigenspace becomes an eigenspace is just the trivial one.
for example, consider the linear transformation defined by then the generalized eigenspace corresponding to is the eigenspace corresponding to