Will the scan gauge mileage functions work when the fuel sending unit is bad?

I believe so. The scangauge works off of how much fuel is being burned, not the sending unit.

Correct. Has no tie to the actual fuel gauge sender. But for it to work correctly you must fill the tank every time. I used mine for 3 months to gauge fuel because my fuel gauge was funky too. Once its calibrated leave the ScanGauge II alone during your fill ups. You'll find going from station to station and pump to pump the gallons number will vary.

Do you mean I shouldn't be using the "fill up" function?

I never partially fill the tank, and I always select "fill up" and hand calculate mileage to compare accuracy. (It's pretty dang accurate every time)

Nope use the functions. The amount of fuel burned should be accurate with how much you put in.