If we renew a home equity loan for a different amount (say less than the original amount) is a new mortgage note required?

This really depends on the circumstances. If the term is up, you probably need a new note. If you are changing the loan terms (interest rate, etc.), but the term of the loan still has some time to go, a loan modification agreement may work. You need new disclosures for a new note. You may need new disclosures for a loan modification agreement, depending on what terms are changed. You may also need a mortgage amendment, depending on whether the mortgage states that it secures replacement notes and modifications of the note.