I don't quite know what your main question is asking, or what (3) means, but (1) and (2) seem to have straightforward answers:

• Any nontrivial forcing makes $L$ non-E-r.e.: if $V\models ZFC$ and $V[G]$ is a nontrivial forcing extension of $V$, then $V[G]\models V\not=L$, and hence $V[G]$ thinks $L$ is not $E$-r.e.

• And there are only countably many E-r.e. sets (every E-r.e. set is the domain of an E-recursive function, and there are only countably many indices for such), so even if $V=L$, there are constructible, non-E-r.e. sets - indeed, even sets of naturals. So I'm not sure if this was the question you meant to ask.

I don't quite know what your main question is asking, or what (3) means, but (1) and (2) seem to have straightforward answers:

• Any nontrivial forcing makes $L$ non-E-r.e.: if $M\models ZFC$ and $M[G]$ is a nontrivial forcing extension of $V$, then $M[G]\models \neg\mathsf{(V=L)}$, and so (from the perspective of $M[G]$) we have that $L$ is not $E$-r.e. (I've edited this bit to avoid some unfortunate abuse of the letter "V.")

• And there are only countably many E-r.e. sets (every E-r.e. set is the domain of an E-recursive function, and there are only countably many indices for such), so even within $L$ there are constructible, non-E-r.e. sets - indeed, even sets of naturals. So I'm not sure if this was the question you meant to ask.