Let's scratch (2) because its too hard. But for (1) I was hoping for either an exact expression OR some approximate expression that works well for strings with $L \geq 10$ and $q \geq 2$ or so. I'm less interested in asymptotics, but that's not to say I didn't appreciate the paper.

From the paper (researchgate.net/publication/…) I don't think there's going to be a reasonable analytic expression for the exact probability for (1). Scenario (2) seems hopeless. Do you agree with this?

@BrendanMcKay Sorry for the typo, it should now be fixed. Also, thanks in general for your comment - this ended up being tricky for me to work out and I feel relieved to know at least one other person doesn't think the problem is trivial.