The card types in Magic are artifact, creature, enchantment, instant, land, planeswalker, sorcery, and tribal (a card type that appears on some older cards). Supertypes (such as legendary and basic) and subtypes (such as Human and Equipment) are not counted.
Most triggered delirium abilities use an intervening “if” clause. There must be four or more card types among cards in your graveyard in order for these abilities to trigger, otherwise they never trigger at all. There’s no way to have the ability trigger if there aren’t enough card types, even if you intend to raise that number in response to the triggered ability. The number of card types is checked again as the trigger resolves, and if it has become too low somehow, the ability does nothing. If which card types are in your graveyard changes but the quantity of card types stays the same (or increases), then the delirium triggered ability will still resolve.
If Angel of Deliverance deals lethal damage to a creature, that creature can’t be targeted by Angel of Deliverance’s triggered ability. It’s already in its owner’s graveyard.
If Angel of Deliverance deals damage to multiple creatures, players, or planeswalkers at once (for example, if it’s blocked by two smaller creatures) its last ability triggers only once.
Because you consider only the characteristics of a double-faced card’s front face while it’s not on the battlefield, the types of its back face won’t be counted for delirium.