> As I understand it, a zombie process has died but still exists as a placeholder in the process table until its parent (or init if the zombie is itself an orphan) checks its exit status.
Correct.
> And my understanding of orphan processes is they are processes that are still alive and running but whose parent has died.
Correct.
> Since a zombie is already dead, its children would be considered orphans, wouldn't they?
Yes. When the parent dies, it's dead. With respect to its children, it doesn't matter whether the parent stays on as a zombie: the children become orphans at the time the parent dies, and then they lose any connection with their parent.
> Would they be affected be reaping the zombie? Specifically, would init adopt them as its children only once the zombie was reaped, or would they be adopted as soon as the parent became a zombie?
No, and the latter, as per above.