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Secondly, this is, indeed, the "Three Step Refinancing System." The new interest rate is 3.625% versus the current 4.875%, resulting in a P&I decrease of $245.61. I understand the bank may get the $ at 3.5% and profit in that way.

No, that's typically not the way it works. The BANK doesn't make money on the YIELD (interest rate). The INVESTOR makes money on the yield. The bank makes money on buying a sack of money at X and re-selling it at Y. That's call loan origination.

Here's an article that 'splains--keeping in mind that very few banks own the loans they originated.
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