A: No. You must pay the cost fully before you can trigger the effect. If the cost is to take a damage but you prevent the damage, then you have not paid the cost and therefore cannot resolve the effect. -(Developer Ruling, Hall of Heroes)
Hero Action: Exhaust Focused Rage and take 1 damage → draw 1 card.
Mulligan for this card. No matter what aspect you are playing, a She-Hulk with Focused Rage installed is just better than one without it.
Get more oomph out of a Legal Practice play you're planning that round.
Incrementally charge up your Gamma Slam.
Reliable insurance all game against those "If I just had ONE more resource/card" hands.
A three-drop is expensive early on, yes. But this pays dividends all game and is one you want in your opener, no question.