Flip-flop hubs are a thing:

https://cyclistzone.com/what-is-a-flip-flop-hub/

In my case (scroll back 35 years), I would ride a fixed gear around the city, but change to the single-speed freewheel for Toronto's legendary "Doughnut Ride" that left Laird & Eglinton every weekend rain or shine, snow, sleet, or hail.

In very olden times, riders would dismount and flip their wheels around in the middle of a Grand Tour before climbing the big mountain. But in contemporary riding, flip-flop hubs are just a convenience of sorts that make it easy to decide what gear you want to use for each ride, I have never flipped the wheel in the middle of a ride.

p.s. I still have one bike with a flip-flop hub: It's a 24" trials bike, and while I run an 18t freewheel on the cranks, the flip-flop lets me choose between a 15t for pure trials work or a 12t for goofing around and having a tall enough gear to ride places, like on my local double black-diamond MTB trails.

Yup (to the first). Fixie riders can achieve two 'gears' by mounting a sprocket on both sides of the rear wheel and flipping the wheel round if needed. More, as always, from Sheldon Brown: https://www.sheldonbrown.com/fixed-conversion.html#flip-flop

If only there was an invention that enabled you to change gears on the fly with the push of a stalk on the handle bars…

First one is correct, you can even have a fixed / freewheel combo with one on each side. Gears have to be close in size to allow chain slack to be taken up by the axle slots. Second one would only work on a few bicycles as generally the front hub is narrower tan the rear.

It's a pretty common thing called a "flip-flop hub."

https://en.wikipedia.org/wiki/Flip-flop_hub