Are institutions required by regulation to retain the original executed mortgage and note in paper form?

For what purpose? Electronic records are State law issue.

It was just a question that came up....I have always understood that the original signed mortgage and note could be converted into electronic form with no problem, but was challenged on it. Would the secondary market require it?

It depends, but there are requirements for originals for FNMA, etc. depending on the circumstances.

If you pledge loans to Federal Home Loan Bank they require keeping them.

Once the loan is paid in full, if you do not have the original recorded mortgage, you will need to prepare a separate discharge of mortgage form in order to cancel the mortgage of record (at least, in New Jersey you will; don't know about other states).

I second the Federal Home Loan Bank requiring originals. We've been told to keep originals for that reason by our general counsel.