Can I use a card with bloodrush in response to something?
Yes. All activated abilities are at instant speed unless they specifically indicate otherwise*. Because the ability is written with a colon, it is an activated ability. It does not need to be on the battlefield to be activated, and your opponent does not need to know it exists to activate it (although you must reveal it, obviously). In this particular cast, you're discarding the card as part of the cost. That's the only reason you have to discard the card. If it didn't say discard, you could activate the ability as many times as you wanted.
This is spelled out in 112.3b, emphasis mine.
Compare this to the rules text for instants,
So the timing for activated abilities is the same as the timing for instants.
As for it being in your hand, this is spelled out here:
Note: I said specifically indicate otherwise. This specification does not need to be on the card. For example, the Comprehensive Rules state that planeswalker activate abilities can only be activated on your turn at sorcery speed, but this is not stated on the card.
Yes, your Bloodrush would stack on top of Izzit Charm and thus resolve before the damage. Bloodrush was specifically designed to be used in the combat phase, at instant speed and only on target attacking creatures.
Also note that because this is an activated ability, this should go without saying but is still questioned again and again, cannot be countered by counter-spells. It can however be countered by the following only because Activated Abilities are named.
"Bloodrush" has no rules meaning (as indicated by the use of italics), but it's only used on activated abilities (that can be activated from your hand), and you can activate activated abilities whenever you have priority ("instant speed").