When mailing out checking account statements, are we required to also send the images of the checks?

That is a state law issue. The "standard" version of the UCC doesn't require it, but your state might have tweaked its version of the UCC in some way.

Do banks typically include check images with Money Market and HSA accounts? We include check (paper debit) images with all accounts but will stop including with Savings Accounts effective 1/1/17.

Check the Texas UCC. It has requirements for making copies of checks available on request.