Quantum-Cosmic Jump: Once per game, give Keeper a power action if he occupies your starting area. Place him in any square of the map that's not in or adjacent to a starting area used this game.

To Feed the Hunger of Galactus: Give Keeper a power action. Remove an action token from a friendly character within 5 squares and line of fire.

Keen Cosmic Awareness: Keeper can use Outwit, but only to target characters of a lower point value. When he does, if he counters a standard power in this way, while that power is countered no opposing character can use that same power.

Will Power Cosmic or other similar effects protect agains Keen Cosmic Awareness?
Depends on what you mean. Initially, Keeper is using Outwit... just plain ol' Outwit. So, yes, Power Cosmic (and other such things) will prevent Keeper from using Outwit on that character. However, once he has used Outwit on a character, the rest of the power just says that the other opposing characters can't USE that power. Power Cosmic and the like do nothing to prevent that part of this special power.

Do the other opposing characters have to cost less than the Keeper?
No. Keen Cosmic Awareness only specifies a point limitation for the use of Outwit. Once Outwit is used, Keen Cosmic Awarenss doesn't care about the point cost of opposing characters.