If I copy a figure(with an entity attached)using Hope summers' omega level mutant, do I copy whatever the entity gives the possessed char too?

No. The possessed character can only use the power granted på the entity and Hope summer only copies powers that the character possesses.

You forgot to list Hope's SP (for reference):

Yep, Hope doesn't require then to possess powers for her to use them, unlike most power copycats. About the only things she can't copy are non-power granting traits (ie traits that don't say can use power xxx). Long story short, yes she can copy a possesed piece's entity powers (that are currently chosen).

No. She can only copy possessed powers.

Player's guide section 03 page 25:

Ah, thanks for the update, didn't realize they errated it to possesses.

They didn't; they merely clarified the existing wording. "The powers of one chosen adjacent character" means the same thing as "the powers possessed by one chosen adjacent character."