Why birds do not fall while sleeping

Kangaroos?

Also, I see both of my dogs standing on 2 legs every day, often walking short distances like that. According to wikipedia this only happens when they are trained to do it (?!) but we never trained them and they've been doing it since a few months old. Maybe I should update https://en.wikipedia.org/wiki/Bipedalism to indicate training may not be required for temporary bipedal behavior in some dogs.

To answer the question: a yacht race?

When birds relax, their feet curl in a similar way, and this provides enough grip that if they're on a perch, they will automatically hold on as easily as if they're awake.

There's a bit more to it, as the way the tendons in their feet and legs are attached, the foot will automatically grasp when the ankle is bent, so it's a much stronger grip than our floppy relaxed fingers would provide. Here's a quick rundown with a good gif illustrating how it works. https://windycityparrot.com/birds-sleep-standing-one-leg/ They also have an extra balance organ between their hips that help them sray upright, so the whole anatomy lends itself to sleeping like this.

Additionally, when we're awake and moving, we're constantly on our feet, so our feet and legs will get tired and need a rest. However, our arms don't generally get tired just from walking around, right?

Birds are the opposite. They spend a lot of time flying, and their feet are mostly relaxed and resting while they're in the air. It's the wings and chest muscles that get tired.

So sleeping on their feet has a whole other connotation to them.