regarding your second question, i'm not sure what you mean by a "full rank eigenspace". so i'll wait until you define that term.

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