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You can settle the matter before OR after the garage carries out the work - there is no fixed rule. Either way, you need to agree a figure with the other party and make a payment which is in FULL AND FINAL SETTLEMENT. I recommend you contact the other party and get a final figure in writing; then send the payment you agree by cheque or bank transfer, so you have evidence of payment, with a covering letter sent by recorded delivery saying the payment is in full and final settlement. Ideally you should get the other party to send a receipt.
I'm not sure why the police have given you that information because strictly speaking the owner of the car is the person who sustained the loss/damage and he/she is the person who is entitled to be compensated. Paying the garage directly means that you have some proof that the repairs have been paid for this cutting out the 'middleman' but there is nothing wrong with paying the owner directly. So long as you have proof of payment in full and final settlement then it doesn't matter whether you give the money to the car owner or garage - it still amounts to the same thing.
No, that's not the issue. The issue is that his car was damaged and he is entitled to be compensated. Whether he gets the vehicle repaired or not is up to him. What you are paying the other person for is damaging his property. How and when and where he gets the damage repaired is entirely up to him. There is nothing in law that says he must get the car repaired; but the law does say that if you damage somebody else's property you have to compensate them.