If the equipped creature is dealt lethal damage at the same time that you gain life, it will die before Excalibur II's first ability would resolve.

Each creature with lifelink dealing combat damage causes a separate life-gaining event. For example, if two creatures you control with lifelink deal combat damage at the same time, Excalibur II's ability will trigger twice. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, planeswalkers, and/or battles at the same time (perhaps because it has trample or was blocked by more than one creature), the ability will trigger only once.

If you gain an amount of life "for each" of something or "equal to the number" of something, that life is gained as one event and Excalibur II's first ability triggers only once.

In a Two-Headed Giant game, life gained by your teammate won't cause Excalibur II's first ability to trigger, even though it caused your team's life total to increase.

Excalibur II's first ability triggers just once for each life-gaining event, whether it's 1 life from Al Bhed Salvagers or 3 life from Balamb T-Rexaur.