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would using psychic powers that work like firing a weapon count as using a weapon?
Like, can a librarian use an offensive psychic power and shoot with his bolt pistol in the same turn? (since psychic phase and shooting phase are now different)
With that, can you use as many psychic powers as you want the same turn as long as it matches your mastery? (like using 3 lvl1 powers in one turn if your mastery is 3)

You can cast and shoot at different units, with tge psychic phase there is no conflict anymore.

BRB you could cast until you run out of warp charges, but the FAQ changed that to as many powers as the casters mastery level. The warp charge requirement is irrelevant, ML3 = 3 powers.

^ too add on from he said

You can also shoot in the shooting phase. None of the psychic phase effects your shooting, unless through exceptions (Like moving or 'counts as moving' when using some powers)

You can even use more than one Witchfire power, and target a different unit with each. See page 27, "Witchfire" second paragraph.