You're right on both accounts.
1. First notice that if $p(x)$ has zero constant coefficient, say $p(x)=x^mq(x)$, then $p(x)$ is a zero divisor if and only if $q(x)$ is.
So suppose $p(x) = \sum_0^n a_ix^i \in \mathbb{Z}_4[x]$ has an odd coefficient and a nonzero constant coefficient, and let $k \le n$ be least such that $a_k$ is odd. Then all the $a_j$ for $j < k$ are even. If $k=0$ then clearly $p$ is not a zero divisor (why?) so we must have $a_0=2$.
If $q(x) = \sum_0^m b_ix^i \in \mathbb{Z}_4[x]$ is another polynomial (with nonzero constant coefficient, for the same reason as above), then the coefficient of $x^k$ in $p(x)q(x)$ is
$$\sum_{i+j=k} a_ib_j$$
If $p(x)q(x)=0$ then we must have $b_0=2$. But then
$$\sum_{i+j=k} a_ib_j = 2a_k + \cdots \equiv 2 + \cdots \pmod 4$$
and so $b_{\ell}$ is odd for some $\ell<k$; otherwise the other terms would all disappear and we'd be left with $2$.
But $q$ is a zero divisor with nonzero constant coefficient, and so interchanging the roles of $p$ and $q$ in the above, we see that $a_i$ is odd for some $i<\ell<k$, contradicting minimality of $k$.
2. Certainly if $p(x)q(x)=1$ then their constant coefficients are either both $1$ or both $3$. So $p(x)=1+g(x)$ and $q(x)=1+h(x)$ for some $g,h$ with equal even constant coefficients; and then $g(x)+h(x)+g(x)h(x)=0$.
Let $g(x)=\sum_0^n c_ix^i$ and $h(x)=\sum_0^m d_ix^i$, and let $k$ be least such that at least one of $c_k$ or $d_k$ is odd. [We must have $k \ge 1$.] Then
$$c_k + d_k + \sum_{i+j=k} c_id_j = 0$$
We must therefore have $c_k$ and $d_k$ both odd. But $c_0=d_0$ is even, and so
$$c_k + d_k + \sum_{i+j=k} c_id_j = c_k+d_k+c_0d_k+c_kd_0 = (1+c_0)(c_k+d_k) \equiv 2 \pmod 4$$
since the rest of the terms in the sum are zero.
This is a contradiction, so $g$ and $h$ must both be divisible by $2$, and so indeed $p(x)=1+g(x)$ where $g$ is a zero divisor.
