Has this model of random directed graphs been studied?
It is very easy to say something about the expected length before you get stuck in a loop: if there are $n$ videos, it will (starting from a random video) take in expectation $\Theta(\sqrt{n})$ videos before you loop around (the actual value is around $1.25\sqrt{n}$). This is effectively the birthday problem, since each time you draw a video at random.

Since each video in the chain of $\Theta(\sqrt{n})$ videos is equally likely to be the one you loop back to, the average length of a loop is also $\Theta(\sqrt{n})$ videos (actual value $0.625\sqrt{n}$).

This gives the expected length of the loop you end up in after starting from a random video. This means that a loop with many videos leading to it is counted more strongly. If instead you want to know the expected length of a loop if you pick a random loop, this may be found as $T(1)$, where

$$T(i)=\frac{n-i}{n}T(i+1)+\frac{i}{n}\frac{i+1}{2}$$

and $T(n)=\frac{n+1}{2}$. Computing the values of $T$ experimentally it seems to match up with $0.625\sqrt{n}$, so both ways of counting the expected loop length are the same.

Computing the expected number of cycles seems to be a harder problem. We start by counting the expected number of cycles of a given length. The probably that a node is part of a length-1 cycle is $\frac{1}{n}$, so there are in expectation $1$ length-1 cycles. The probability that a node is in a length-2 cycle is $\frac{n-1}{n}\frac{1}{n}$, so there are in expectation $\frac{1}{2}\frac{n-1}{n}$ length-2 cycles. In general, the number of cycles of length $l$ is $\frac{1}{l}\Pi_{i=1}^{l-1} \frac{n-i}{n}$.

We can obtain a crude upper bound on the number of cycles by considering $\Sigma_{i=1}^n \frac{1}{i} = H_n$ which is $O(\log n)$. Unfortunately the number of cycles doesn't seem to converge to $\log n$ so this bound is not tight.