How can we know if an extension of $\sf ZF$ by some large cardinal property that results in a consistency strength beyond $0^{\#}$ is compatible with choice or not?
I mean the easiest way to know if an extension of $\sf ZF$ is compatible with choice is to construct a model of it in $L$. But if any extension goes beyond $0^{\#}$ then no model of it can be constructed in $L$, and so there is no uniform way to prove its compatibility with choice.
Is there something that takes the place of $L$ for those high kinds of extending $\sf ZF$? Is it $\sf HOD$? Or perhaps other known models in which choice hold. Also, are there known models that can interpret partial forms of choice, like $\sf CC$ and $\sf DC$ principles at such altitudes?
This is the territory of inner model theory.
The issue with $\mathsf{HOD}$ is that it's far too flexible: there's generally no easy way to argue (and indeed it's usually false) that passing to $\mathsf{HOD}$ preserves large cardinals. So the mere fact that $\mathsf{HOD}^\mathcal{M}\models\mathsf{ZFC}$ whenever $\mathcal{M}\models\mathsf{ZF}$ doesn't help us much. Instead, we need a more explicit construction. This is where mice and their ilk enter the picture (Schimmerling's The ABCs of mice is the closest thing I know to a readable intro to mouse theory).
However, it's also worth noting that the definitions of large cardinal notions often rely on choice in subtle ways. For example, it's (we hope!) consistent with $\mathsf{ZF}$ that $\omega_1$ is measurable via the club filter. Similarly, we currently don't know whether "If a Reinhardt is consistent with $\mathsf{ZF}$ then it's consistent with $\mathsf{ZFC}$" is correct (we know the conclusion is false but the hypothesis could be true). And speaking only for myself, I am not entirely sure what the "right" choice-free definition of a supercompact cardinal is in the first place! So you should absolutely not think of relative-choice-consistency as a trivial or automatic sort of result, once we climb up to the exciting levels of the large cardinal hierarchy.
That said, a good starting point is definitely the model $L[\mathcal{U}]$, the "measurable-accommodating" analogue of $L$. If memory serves, both Jech and Kanamori have good treatments.