Double and triple checks must be made in 1 move. If a player moves a piece that allows a discovered attack on another player's king and at the same time checks another king (be it by discovery or standard check), that is a double check (even if it wasn't his piece). The same applies for triple checks.
So to answer your questions:
A standard double check against one king does not count. You need to check 2 kings.
You will get the points if you create a discovered attack using an opponents piece that causes check
You can deliver a check against a king which is already checked by another player
It does not matter if an opponents king is checked before you move. It is only checks you create with the piece you move and discovered checks created by the piece you move that counts and it must be to different kings.
There are some special cases that I cant understand.
It is obvious that a check delivered to two Kings in one move is a double check. The same to three King is a triple check. But...
Is
I just want to know it so I can collect points when I am seeing opportunities.