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What is the number of permutations $piin S_n$ that are products of disjoint cycles of distinct length? What is the number of permutations that are products of disjoint cycles such that no more than $k$ cycles are of any given length?

I am really interested in the asymptotics of the proportion of permutations in $S_n$ with these properties as $nto infty$. What is a good reference for this sort of statistics?

If we denote by $c_i(sigma)$ the number of cycles of length $i$ in $sigma$, we can write the exponential generating function of permutations with cycle statistics as
$$sum_ngeq 1sum_sigmain S_nleft(fracx^nn!prod_igeq 1t_i^c_i(sigma)right)=expleft(sum_igeq 1 fract_ix^iiright) =prod_igeq 1 left(1+fract_ix^ii+fract_i^2x^2i2i^2+cdotsright)$$
From here we see that the exponential generating function of permutations with distinct cycle sizes can be obtained by removing all terms where any $t_i$ has exponent $geq 2$, and then setting all $t_i=1$. So we get
$$prod_igeq 1left(1+fracx^iiright)$$
From here the methods of A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics by P. Flajolet, E. Fusy, X. Gourdon, D. Panario, N. Pouyanne show that the coefficient of $x^n$ is asymptotically equal to
$$e^-gamma+frace^-gamman+Oleft(fraclog nn^2right)$$
which means that our desired number of permutations is asymptotically given by $$n!left(e^-gamma+frace^-gamman+Oleft(fraclog nn^2right)right).$$
Notice that in section 3 they actually provide much more refined asymptotics, in case you wanted more terms. Moreover, I believe their method should let you compute the asymptotics for permutations with cycles of the same length appearing at most $k$ times whose generating function is given by
$$prod_igeq 1left(1+fracx^ii+fracx^2i2i^2+cdots +fracx^kik!i^kright)$$
from the same considerations as above.