Yes and no to some of this -- here's a more clear explanation (if you found any specific explanation confusing, feel free to point it out so it can be edited for clarity!).

Characters are given one magical ability as part of the game mechanic. This power is never affected by any world frequency -- ICly it's given so that characters will always have a power, regardless of the frequency of the world. This power will always be with the character and is completely up to player discretion.

Powers and magic that characters have within their own canon (or in the case of a CRAU, because of another game's mechanic) will fluctuate depending on the frequency of a world. On the game hub, these powers will never be usable. Depending on the world that is traveled to, these powers may be usable.

(This is speaking in general terms; some events may play with this mechanic, ie. powers going haywire and somehow being usable at the hub.)

As an added note, the given game power DOES stay when characters are on high frequency worlds. So if you give your character laser vision and in canon they have telekinesis, they would have both laser vision and telekinesis on worlds with high frequencies.

Hopefully that's more clear! The FAQ and other pages will be combed through to make sure this is more apparent.