Let $F = F_k$ be a free group of finite rank $k$. The length of a conjugacy class is defined to be the word-length of a shortest representative.
Given a number $n$, how many conjugacy classes of length $n$ are there?
A precise formula would be great, but even an asymptotic estimation would be enough (even only for $F_2$).
More generally, is there any reference that studies combinatorial properties of free groups and their conjugacy classes?
This is exponential. Indeed, a lower bound on the number of conjugacy classes of length $n$ in the free group of rank $k$ is: $$2k(2k-1)^{n-2}(2k-2)/n.$$ Here, the $2k(2k-1)^{n-2}(2k-2)$ is the number of freely and cyclically reduced words of length $n$, whilst dividing by $n$ partitions these off into conjugacy classes. Its a lower bound because we don't always need to divide by $n$ (consider, for example, the word $a^n$). The exponential term $(2k-1)^{n-2}$ dominates as $n\rightarrow\infty$.
The key phrase you can google is "conjugacy growth function". From memory, this topic was first formalized and studied in this paper of Guba and Sapir, so you can look at the papers which cite it. [Conjugacy growth typically refers to the area of balls, and you care about the perimeter, but they are related.]