Definition of support in the context of an encryption scheme and usage of P = NP assumption in lemma 9.2 (Arora, Barak)
If you read the statements carefully, you see that $E_{U_n}(0^m)$ does not denote the encryption of $0^m$ with a specific key. It is function applied to a probability distribution - so $y$ is actually a probability distribution itself.

And this means, the word support fits to your understanding: It is the minimal set in the codomain (seeing $E$ as a function), where the probability is not 0 for any input from $U_n$ (side-note: the term closed doesn't really do anything for discrete or finite sets) - it is precisely the image of $E$ for this specific key length.

Another way to see this: The set $S$ is the image of the encryption function - for the specific parameters $m,n$.

Regarding this statement:


  Note that $y \in S$ if and only if $y = E_k(0^m)$; hence if
  $\boldsymbol{\mathbf{P}} = \boldsymbol{\mathbf{NP}}$
  then membership in $S$ can be efficiently verified.


This sentence might be missing something: It should state for claraification, that this is true for some key $k$ taken from $U_n$. 

Regarding your 2nd question: For any $NP$ problem, verifying a witness if polynomial. But given some value in the codomain of $E$, you can't decide if it belongs to $S$ or not. However, if $P=NP$ you can find a witness in polynomial time. If for some potential witness exists, it is in $S$, and if no witness exists, it is not in $S$.