You still pay the mana cost when casting the exiled card, and you must follow all timing rules for it.

If an instant or sorcery spell you control with lifelink deals damage to you, you gain and lose that much life simultaneously. In this case, your life total doesn’t change. (You may gain life if the spell also deals damage to other permanents and/or players, however.)

Some instant and sorcery spells instruct other things to deal damage rather than deal damage themselves. In these cases, you won’t gain life for that damage (unless the other things also have lifelink). Notably, a spell that instructs a creature you control to fight won’t cause you to gain life.