Bloodhall Priest

If you have a card in hand at the moment the trigger condition occurs, Bloodhall Priest’s ability won’t trigger, even if you can get that card out of your hand immediately afterward. If you have a card in hand as the ability resolves, it has no effect. No damage will be dealt.

A card with madness that's discarded counts as having been discarded even though it's put into exile rather than a graveyard. If it was discarded to pay a cost, that cost is still paid. Abilities that trigger when a card is discarded will still trigger.

If you discard a card with madness to pay the cost of a spell or activated ability, that card's madness triggered ability (and the spell that card becomes, if you choose to cast it) will resolve before the spell or ability the discard paid for.