What happens to the $ when a points plus $ award is redeposited?
 

BA gives you the option to redeem points plus dollars in addition to just points for a flight, i.e. I have the option of paying 200,000 points plus $891 or 96,000 points plus $3531. If I go with the second option, then cancel and redeposit the award, am I refunded 96,000 avios and $3531-redeposit fees, or 200,000 avios plus $891-redeposit fees?

It all gets refunded. There is further information in this thread.