Patrick,
Ditto here, or happy to come out with you to demonstrate how it is done.
One thing I never do is attach the warp to the stern of the boat as if stuck it is surely the quickest way to pull the boat down. Also, if done properly the warp should never be under the boat.
Regarding which side, I steam my anchor out with the buoy on the starboard side, for no other reason than the helm is on that side, so easy to keep an eye on the buoy at all times.