It seems to me that radicals, natural numbers without power factors, generalize the concept of primes. You could ask after the nth radical and the number of radicals less than a specified number. But no one seems to do that.
About 60 % of all integers, in all intervals I have tested, seems to be radicals and the function $r_n$ (nth radical) seems to have an "asymptotic" line $\approx 1.64\cdot n-1,15$ with very small but irregular variations around that line.
A monoid structure is defined by $\operatorname{rad}(a\cdot b)$ on the set $\mathcal R$ of all radicals and any prime can be estimated by radicals $r_1<p<r_2$, as it seems so that $r_2-r_1\lesssim 4\cdot\log_{10} p$ .
My question is: are there elementary theories about radicals as such? Except from the ABC conjecture, are there other conjectures?
What you call radicals are called square free numbers. Some of your questions have been studied.