Mortgage Payoff Interest after Refinance

The new mortgage company gets the current balance from the existing mortgage company, and calculates any accrued interest that will need to be paid off. They probably got that balance well before the closing date, and were calculating more than a month's worth of accrued interest (and didn't account for a payment being made in between). So it's a "conservative" amount from their point of view to make certain that the old loan is paid off completely (and the lien can be released).

I also recently refinanced, and the "payoff amount" was much higher than the balance at the time plus accrued interest according to my calculations. The explanation that I got was that it included a "buffer" in case there were any delays in payment, discrepancies in the escrow account (even though I don't use an escrow account) or other unanticipated differences, but that my current mortgage company would refund me the difference. Sure enough, after about a week I received an "escrow settlement" for the difference. Other than having to have a little more cash at closing, everything worked out in the end.

You have probably also made a payment since the payoff figure was calculated, so your actual payoff will be even less if they did not take that into account.

Any interest between the date of closing to the start of the month before your first payment should be included in the closing settlement as "prepaid interest" and calculated at the new rate of interest (since that's the rate of the loan you're taking out). So yes, your understanding is correct.