[color=red]1-He rolled up the window of the car on the driver’s side.
[color=indigo]2-He rolled up the window of the car that was on the driver’s side.
[color=red]
3-I greeted the brother of the sheriff with a broken leg.
[color=indigo]4-I greeted the brother of the sheriff who had a broken leg.
(I greeted that brother of the sheriff who had a broken leg. The sheriff’s brother has a broken leg.)
I think 3 and 4 would more readily mean that the sheriff had a broken leg.
1 / 2 It sounds as if the car was on the driver’s side of something else. Try this:
He rolled up the window (that was) on the driver’s side of the car.
Alternatively:
He rolled up the car window (that was) on the driver’s side.
He rolled up the driver’s side window of the car.
3 / 4 The same problem exists and as you say, the indication is the sheriff has the broken leg.
This time only the alternative option works well because the context has changed:
I greeted the sheriff’s brother who had a broken leg.
You cannot use ‘with’ - it indicates that the means of greeting you used was the broken leg!