Regarding present continuous questions (see below), why is "doing" a correct form when the sentence is talking about a future event? "doing" is a present continuous verb of "do" which means something happening now. However, the sentence is clearly referring something about future, so why is usage of "doing" permitted? Please explain why, thank you!!
This is present continuous, as you correctly said, but it is present continuous for future use.
You use the same construction as with PC but then add a time in the future.
I am driving to London (i.e. I am in the car).
I am driving to London this evening/ tomorrow/ at the weekend.
(I have made arrangements to drive to London but Iam not yet in the car).
So, for your example:
What are you doing? (PC)
What are you doing this evening? (PCFU)
It is a Simple Future, often called the Diary Future.