Verifiers equivalent classes
To your first question, suppose your non deterministic machine has two transition functions $\delta_0,\delta_1$ (if you allow arbitrary number of transitions, then you can construct an equivalent Turing machine with only two transitions by adding more states). In that case, it is intuitive how to use the one time read witness. Read $w\in \{0,1\}^{p(|x|)}$ as a computation sequence, i.e. the $i'th$ bit is $0$ if $\delta_0$ is used and $1$ otherwise. Follow the computation described by $w$, and accept iff the original non deterministic machine accepted in this computation. I leave it to you to complete the details.

As for your second question, the class generated is indeed $NP$. Obviously $VNL\subseteq NP$, so it remains to see why $NP\subseteq VNL$. Let $L$ be some language in $NP$, so we have $L\le_p SAT$, where the reduction can be computed in logarithmic space (verify that the reduction used in the proof of Cook-Levin's theorem can indeed be computed in logspace). Let $f$ be the reduction from $L$ to $SAT$ computable in logarithmic space. Given $x\in \Sigma^*$, we would like our witness $w$ to be a satisfying assignment for $f(x)$, since we can verfiy an assignment in logarithmic space. The problem is that $|f(x)|$ is polynomial in $|x|$, so we cant write it in our work tape. You can solve this by having $f(x)$ provided in the witness (along with an assignment), and your machine could verify for all $1\le i\le |f(x)|$ that $w_i = {f(x)}_i$. Luckily, since $f$ is computable in logspace, you can verify that the witness provided the correct $f(x)$, and check if the assignment satisfies $f(x)$.