I have lived in two places with breakfast bars in American style kitchens. I found that I tended to spend a great deal of time sitting there so a bar stool with a comfortable back is important. I made mine from pine, copied from some I saw in a bar. I was also given an old captains swivel seat from a yacht so I built a stand for it, reupholstered it and unfortunately left it for the new tenant who liked it. That was one of my favourite seats, to sit in comfort at a breakfast bar rather than balancing without a back to lean on.

If you want the seats to swing out from the island be sure to check that the island is substantial enough to take the load, also that the island is well anchored to the floor. If you consider the arm is for example 2 feet long, then sitting on the extende seat will be the same as applying your full weight to a crowbar, two people sitting on one side would destroy a normally constructed island. Similarly, if fitted to a wall make sure that they are anchored well and that the wall structure is suitable. A nice idea to save space though

The swivel ones usually have a lever that activates the gas that allows it to go up and down. The gas mechanism is usually within the middle pole. That way you can adjust the height. I have seen some that adjust the height by swivelling round, swivel one way the seat goes up and swivel the other way it goes down.

Google says potterbarn.com do, but not here in the UK. Houzz shop is just links to other stores.