The question in your motivation is much easier to answer than the actual question! But first: the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_q[x]$ is
$\frac{1}{n} \sum_{d | n} \mu(d) q^{n/d} = \frac{1}{n} \left( q^n + O(\sqrt{n} q^{n/2}) \right)$
by Möbius inversion, from which it follows that for fixed $n$ and $q \to \infty$ the probability that a monic polynomial of degree $n$ over $\mathbb{F}_q[x]$ is irreducible is $\frac{1}{n} + O(\sqrt{n} q^{-n/2})$. In particular it approaches $\frac{1}{n}$ exponentially fast.
If $p$ is lifted to a polynomial $\tilde{p} \in \mathbb{Z}[x]$, for example by identifying $\mathbb{F}_q$ with $\{ 0, 1, ... q-1 \}$, then if $p$ is irreducible, so is $\tilde{p}$. It follows that the probability that $\tilde{p}$ is irreducible is at least the probability that $p$ is irreducible.
To show strict inequality it suffices to exhibit a monic integer polynomial of degree $n$ which is irreducible but reducible $\bmod q$. There are many ways to do this (for $n \ge 2$ of course). For example, if $q \ge 5$ then $p(x) = x^n + (q - 2) x^{n-1} + 1$ is irreducible by Perron's criterion, but $p(1) = q$. If $q \ge 3$ and $\ell$ is a prime dividing $q - 1$, then $p(x) = x^n + \sum_{i=1}^{n-1} c_i \ell x^i + \ell$ is irreducible by Eisenstein's criterion, but by choosing the $e_i$ appropriately we can arrange to have $p(1) = q$ again. If $q = 2$ then I think one can use complex-analytic tricks but I don't know any good way to generate examples off the top of my head.
There are MO discussions about variants of your second question here and here. This is a hard circle of questions, and I am pretty sure the answer to the second problem as written depends strongly on how you identify elements of $\mathbb{F}_q$ with integers.