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Consider the following scenario.

A squad of grey knights charge some necrons and a necron lord.

The justicar issues a challenge to the necron lord.

Now what happens to the psychic powers of the GK squad. The body of the squad wants to cast hammerhand to better kill the necrons but the justicar wants to use his force weapon to better kill the multi wound lord.

Can they both (gk squad and justicar) cast different powers meaning the justicar is separated from the squad when doing the challenge.

No. Brotherhood of Psykers tells you that they act as one psyker.

Just because their attacks and such are separate, does not separate their psychic powers.

Boy this is a good question. Without pouring through the GK codex, I can't be sure. But it appears that you'll have to make a choice. I'd go ID for the whole squad IIWY.

The justicar is still part of the squad, he's just treated as separate for the purposes of the melee. The effect continues.

Ok, how about librarians of any type. lets say a librarian or sorceror issues a challenge or is challenged, now he has powers that both help his squad (in various points in the phase) as well as powers that help him in the challenge. Can he cast both?

He is still part of the squad, so yes.

At no point do models in a challenge cease to be part of their unit.