Havengul Lich's activated ability creates a delayed triggered ability that triggers when you cast the creature card. This triggered ability will go on the stack on top of the creature spell and will resolve before the creature spell will.

If you don't cast the creature card that turn, Havengul Lich won't gain any of its abilities.

Havengul Lich gains the activated abilities of the card while that card is still a spell on the stack.

Havengul Lich's ability changes the zone you are permitted to cast the card from, not the times you are permitted to cast it. You can cast a creature card that has been targeted by Havengul Lich from the graveyard, but unless it has flash, you can only do so if it's your main phase, the stack is empty, and you have priority.

Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keywords are activated abilities and will have colons in their reminder text.