tl;dr: $0 balance on mortgage!
Turns out paying down principal before the new statement with new interest rate does not cause the payment to be amortized with the just-paid-down principal. The monthly payment was calculated with the remaining principal at the time the new interest rate was determined, i.e., 45-days before the change date. So we did not end up with a $49.99/mo mortgage payment. For those looking to pick a desired monthly mortgage payment when an ARM adjusts, you'll need to plan 45 days in advance with some prediction of where interest rates will be.
Instead in December, we paid down enough principal so that the next automatic monthly payment on New Year's Day with the new 3.875% interest rate would exactly pay off the loan. This resulted in 69¢ in interest to be paid at 3.875% instead of the original 2.625% ARM rate (which would have been 47¢ in interest, so we paid an extra 22¢ in interest because of the adjusted higher rate).
Unfortunately, because we paid down the principal so much, the direct debit on our account was automatically cancelled. I only noticed on the 4th as we figured the New Year's holiday and weekend would delay the payment, but it was the 2nd business day after and still no activity. I called the servicer to confirm that there was no direct debit set up anymore and made a manual payment of the monthly payment amount. The next day we received a paper mail saying the direct debit was cancelled due to low balance.
So today, the manual processing of the payment showed an additional 4 days of interest was charged because our payoff payment was made on the 4th instead of the 1st even though a normal monthly mortgage payment would have had a grace period of accepting a monthly payment in the first half of the month without a late fee while still counting as being paid on the 1st. It looks like the servicer was kind enough to adjust the account removing the 9¢ in extra interest from the delayed payoff.
Overall, we paid a little bit over $8k in interest while Zillow estimated value growth is more than 16x of that interest.
Now we wait for confirmation of a mortgage paid off! (Where I assume we'll need to manually file with the county's recorder office and pay that recording fee.)