Is the extension going to be a single or two storey? ie will the fact your neighbour having a window on the side be an issue if it's double storey for light issues?

If not then I don't see why having a diagonal driveway should be an issue. Ideally you would get the kerb lowered on the right side but it's quite expensive to do this.

Surely cheaper to put the extension on the right and leave the garage where it is? That way you don't lose your front garden. You will probably also be required to put in drainage for a new drive to comply with SUDS.

Sorry I may have misunderstood, if your new garage isn’t actually going to be used for the car, then do you need to move the driveway? Wouldn’t the car be fine parked on the drive in front of the new extension? You could just extend the path across the width of the house or on the right hand side for garage access.