What heart pieces are unavoidable?

I'm attempting a three-heart run in The Legend of Zelda: Wind Waker. I'm not picking up the heart containers after bosses, and I'm trying not to get more than three heart pieces.

So far, I've picked up one heart piece, which I believe might not have been avoidable: playing hide-and-seek in Windfall Village, if I'm not mistaken, is required in order to get access to the cabana later and grab the Triforce piece inside of it.

Also, the first time you get a letter in the mailbox, it contains a heart piece. This means that I have to accept the heart piece if I'm going to do any side quests involving the mailbox later on.

1. Am I right in believing that it's impossible to complete the game without getting the first heart piece, or could I have skipped it?
2. If I take both of the heart pieces I mentioned so far, will it be possible for me to finish without being forced to take two more heart pieces? Or are there at least two more main questline events that award me heart pieces?