What happens if Falkenrath Gorger dies while a madness ability it granted is on the stack?
The rules on Falkenrath Gorger already has this specific case covered.


  8/04/2016     If Falkenrath Gorger leaves the battlefield before the madness trigger has resolved for a Vampire card that gained madness with its ability, the madness ability will still let you cast that Vampire card for the appropriate cost even though it no longer has madness.


Since he'll leave the battlefield after you discarded a card the madness will trigger and you will be able to cast your vampire.

Second question: Madness is a trigger, if you choose not to cast the card it'll be put in the graveyard and you can't recast it.


  8/04/2016     If you choose not to cast a card with madness when the madness triggered ability resolves, it’s put into your graveyard. You don’t get another chance to cast it later.