Does Apple change iPhone serial numbers when they refurbish them? Does Apple change the serial number of iPhones they take in when they refurbish them?
Typically no. Serial numbers cannot be 'edited' as such - they can be written to new products (such as a new logic board), but an existing serial on a logic board cannot be rewritten.

However, if a logic board is replaced, then that will result in a new serial number.

As an aside, Apple really does have an excellent refurbishment program. Refurbished Apple products come with a new outer shell, a new battery, and the same full one-year warranty they give to their new products. It is more expensive than buying secondhand, but you're paying for the peace of mind.