If you buy a house on which the seller is willing to carry the note and later, say after 3 years take a mortgage to pay off the seller in full, do you need a whole new down payment? Or does the equity you've already built up nullify the need for a down?

Whenever you need a mortgage, as far as I know you need a down payment. See, in the scenario you described, you haven't built up equity with a bank, but rather with the seller. As far as the bank is concerned, the seller is still paying, because the seller is holding the note. So when you come in, you're still asking for a plain mortgage in their eyes.

However, the down payment is a percentage of the total amount of the mortgage, so if you have been making payments for 3 years, the down payment you need will be much lower.

Depending on your credit and who you are financing through, the required down payment is usually around 20%. VA loans usually don't require any down payment or at the most 3%-5%, and FHA loans are generally around 7% (depends on the house, the location, your credit, etc... .)

An alternative is to get a loan, not a mortgage. This depends on how much you have left to pay off. But if you really can't come up with the down payment, you might see if you can qualify for a regular loan for that amount.