Dawn of Hope's first ability triggers just once for each life-gaining event, whether it's 1 life from Vicious Rumors or 3 life from Creeping Chill.
You can't pay multiple times to draw more than one card each time you resolve Dawn of Hope's triggered ability.
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, Dawn of Hope's ability will trigger twice. However, if a single creature you control with lifelink deals combat damage to multiple creatures, players, and/or planeswalkers 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, that life is gained as one event and Dawn of Hope's first ability triggers only once.
In a Two-Headed Giant game, life gained by your teammate won't cause the ability to trigger, even though it caused your team's life total to increase.
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