So, let's say that I have a Dawn of Hope out. I have 2 Healer's Hawks that attack on the same turn. I pay 2 mana to draw a card for the lifelink from the 1st healer's hawk. Do I need to spend another 2 mana if I want to draw another card or do I just draw another card automatically since there were 2 lifelink events in that turn?
As you stated, there are two lifelink events, both setting up an independent instance of Dawn of Hope's trigger. In order to draw two cards, you have to spend 4 mana; the second trigger doesn't 'remember' you've paid already for the first trigger.
Notice that it says "If you do", not "If you did". It is not checking whether you have, at some point, paid 2 generic mana. It's checking whether you paying 2 happened for that trigger. Every time you gain life, Dawn of Hope puts a "You may pay 2. If you do, draw a card." on the stack. Each time such a trigger resolves, you either pay 2 and draw a card, or nothing happens. If, theoretically, there were a card that says "Every time you gain life, you may pay 2. If you have paid 2 this turn, draw a card." or "You may pay 2. If you do, for the rest of the turn, draw a card every time you gain life." then you could pay just 2 to draw two cards. But this card is worded as "Each time you gain life, you may pay 2 to draw a card", not "You may pay 2 to draw a card every time you gain life".