Removing zombie process from the process table

A C program will end in one of three conditions:

1. The program returns from the main function. If the return value is 0, that indicates success, otherwise the return value is the exit state. C99 also allows (but discourages) a main function with a void return type, where returning from main has the same semantics as returning 0 from a main with int return type.
2. The program returns one of the exit functions: exit, _exit, or the new quick_exit function in C11. These functions all do not return and all have the effect of terminating the program, but what they actually do differs (e.g., quick_exit flushes buffers but does not run honour atexit calls).
3. The OS kills the program in some way, e.g., because the program does something it is not allowed to do.

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.