The Forbidden Names Demand a Blood Sacrifice: Give Baron Mordo a free action and choose a standard power other than a defense power. Baron Mordo can use the chosen power until your next turn. If you do and no opposing character was hit by Baron Mordo, at the end of the turn deal Baron Mordo 1 unavoidable damage.

Ancient One's Best Pupil: Baron Mordo can use Probability Control. If he has one action token, he can use either Outwit or Perplex until your next turn.

With the Ancient One's Best Pupil special power, if I use Outwit or Perplex and then give Baron Mordo a non-free action (resulting in two tokens on Mordo), does the effects of Outwit or Perplex end?
No. The power description tells you he can use Outwit or Perplex until your next turn. So once you actually give him a free action to use one or the other, he uses that power (and maintains its effect) until your next turn. Or until the power is lost or countered, of course.

Can I use both Outwit or Perplex? I know I read something somewhere before about the word "or" not necessarily indicating that you need to choose.
You are correct that "or", by itself, does not necessarily indicate a choice must be made. Emphasis on "by itself".

Sometimes, we will see "once per turn" paired up with an "or"; that is one way of indicating a choice must be made. For Baron Mordo's Ancient One's Best Pupil, we also see the word "either"; "either" and "or" together also indicate that a choice must be made between Outwit and Perplex. Mordo can't choose to use both during any given turn.