For non-owner loans, I don’t think that mtg ins is available at all. You need at least 20% down for a non owner purchase, but 25% down will get you much better pricing.

On any loan, per F/F having 40%+ equity and a middle credit score of 740 or above gets the best pricing.
Having an 800 score and or 70% equity makes no difference.

There is also a pricing hit when doing a “cash out” refi. Paying off a 2nd or HELOC *IS* considered cash out. With a credit score above 680 and at least 40% equity, you may avoid extra costs.

It is very difficult to just refinance an existing 1st and leave the 2nd in place.

Buying a legit 2nd home offers the same pricing as a primary home.

Loan limits are higher than $417K on 2-4 units.
5 units or more is considered commercial financing.
F/F do not buy these loans and none of the above applies to more than 4 units.

I don’t think that you can qualify for a refi if you have had a mtg payment more than 30 days last in the last 12 months.
Mortgage payments that are due on the 1st and incur a late charge on the 16th, are normally NOT considered 30 days late on a credit report until they are paid in the following month.

There is no advantage to paying a standard mortgage payment prior to the 15th. You aren’t saving anything in interest. It really is a grace period. (HELOCS charge daily interest)