A simple clarification on $\#P$ closure properties
Yes, that's exactly what it means. Closure under addition can also be phrased as follows: if $f(x)$ and $g(x)$ are $\#P$ functions, then $h(x)=f(x)+g(x)$ is also a $\#P$ function. As an exercise: How do you construct the machines $N_+$ and $N_\times$? Check that these definitions mean the same. $\#P$ has extraordinary closure properties, for example if $f\in \text{FP}$ and $p\in\#P$ then

$$\sum_{y=0}^{f(x)}p(y)$$

is also a $\#P$ function. We say that $\#P$ is closed under exponential sums.

In general, to say that some set $O$ is closed under an operation $\triangle$, means that if you take $a,b\in O$, then $a\triangle b\in O$. In other words, you can't "get out of $O$" by applying $\triangle$ to elements inside $O$. For example, $NP$ is closed under set intersection, because $L,K\in \text{NP}\implies L\cap K\in\text{NP}$ (why?).