Changing probabilities to 0/1 in definition of class IP
Suppose you have a (randomized) verifier $V$ such that, for all $Q,w$,

$$\begin{align*}
w\in L &amp;\implies \Pr[V\leftrightarrow P\text{ accepts }w]\geq2/3\\
w\notin L &amp;\implies \Pr[V\leftrightarrow Q\text{ accepts }w]= 0.
\end{align*}$$

Since $P$ is one possible value of $Q$, it follows that

$$w\notin L \implies \Pr[V\leftrightarrow P\text{ accepts }w]= 0.$$

This means that for all $w \notin L$, there is no choice of random bits such that $V\leftrightarrow P$ accepts $w$, while for all $w \in L$, there is a choice of random bits such that $V\leftrightarrow P$ accepts $w$.

This gives you a certificate for the claim that $w \in L$: namely, the random bits used during some accepting execution of $V \leftrightarrow P$ on $w$.  By the above arguments, if $w \in L$, such a certificate is guaranteed to exist, whereas if $w \notin L$, no such certificate exists.  $V \leftrightarrow P$ can be used as a verifier to check the certificate.  Since $V \leftrightarrow P$ runs in polynomial time, we have a polynomial-sized certificate and a polynomial-time verifier for $L$, so it follows that $L$ is in NP.

This also helps answer your second question.  If you change the $2/3$ to $1$ but don't change the $1/3$ to $0$, the above argument doesn't go through: you can't conclude that there is no certificate for $w \notin L$, so you no longer have a certificate and verifier for $L$.