Given $d,B>0$ the number of polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ have at least one integer roots should be $B^{O(d)}f(d)$ at some function $f$ (from Random Diophantine polynomials: Percent solvable?). So the probability that a uniformly random polynomial in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$ has at least one integer root is $\frac{B^{O(d)}f(d)}{(2B+1)^{d+1}}\asymp\frac{f(d)}{2B+1}$ which is close to $0$.

- Then what is the average number of integer roots for polynomials in $\mathbb Z[x]$ of degree $d$ and coefficient size at most $B$?

I think it should be $<1$.

However I am not sure of the exact parameterizations.

It might be $\frac1{B^{O(d)}}$ on average.

- What is the average number of integer roots for polynomials in $\mathbb Q[x]$ of degree $d$ and coefficients with numerator of size at most $B$ and denominator of size at most $C$?