You've got most of it spot on. Here's some more.
You do not need a birth certificate. Your passport is the accepted document.
Only for French Polynesia do you need a ticket to return you to your home country - the rest of the S. Pacific accepts your berthing on the boat as your outward passage. However, should you leave one boat for another, you must be sure to be removed from one crew list and placed on the other. - This is to protect both you and the skipper of the boat you arrived on.
I don't remember having to provide proof of sufficient funds until we reached Australia. But I'm not sure.
We were sitting in Colon, Panama when two fellows who were backpacking through Central and South American approached us offering to work as line handlers - they didn't ask to be paid, they just wanted the experience. We took them on, and they were nice fellows. Ran into one of them in the Solomon Islands 4 or 5 years later.
If you can provide proof of some experience, or bring a letter of reference with you, you might find it a bit easier. The neater you are, the better you present yourself, the easier it is going to be. Once you get your first berth, be sure to ask for a letter of recommendation when you leave them. It will make it easier to get the next berth.
Oh, and as you go along, remember that neatness counts. We only took on a hitchhiking crewmember once, but he was the perfect crew. Neat, clean, and he volunteered to wash ALL THE DISHES! You can believe he endeared himself to me. When he came on board, he said to Peter: "You're the captain and I am the crew. I am supposed to do whatever you tell me to do, and anything you don't want to do." With that attitude, that sense of humor, he got off to a good start and we were sorry to see him go on to the next crew position.
Fair winds, and happy holidays to you, too.