Not if the first claim was settled. If you settled, the amount you received was a full and final settlement for all damage caused to your car, and should have been used to complete all repairs.

I doubt it, your car should have been checked out at the time.

There are two points you would have to proove:
1) That the handbrake was damaged in the previous incident.
2) That the damage was not too remote ‘non remota sed proximata causa spectatur’ – doctrine of proximate cause.
I do not see how you could do this. (and this assumes the first claim has not been settled, yet another obstacle)

It would have to be proved that the previous accident caused damage to your handbrake. So you would need to get an independent engineer to look at the car properly. However, your neighbour will be entitled to claim for the damage from your insurance company.