Is getting a random integer even possible?
Most typically you choose a random natural from a finite range. If you toss a fair coin, you get a random number that is either $0$ or $1$. If you call a random number generator on your computer, you might get a number in the range $[0,2^{32}-1]$. There is no problem having equal probability in these cases. If you want to choose a random natural, you can't have the probability of all numbers be equal, but you can use a non-uniform distribution that sums to $1$. If you don't think $0$ is a natural, you can have $p(n)=2^{-n}$, for example.