Why birds do not fall while sleeping

Just as we can still breathe and digest food while sleeping!

We can even adjust our sleeping position while sleeping, possibly to avoid bedsores? But perhaps avoiding falling requires more processing power.

Concurrency at its finest I say!

Yes, adjusting our sleeping position has to do with the sensation of pain, a stimuli (pain, discomfort, pressure points) that cause most individuals to reposition during sleep, i.e. it is the sensation of pain from pressure buildup that triggers the need to move during sleep. You are right, it is to prevent prolonged compression of tissues, to avoid bedsores, nerve damage, etc.

That said, this is not the whole story, because there are people with a rare genetic disorder called CIPA which completely prevents the feeling of pain, but they still reposition themselves, so besides pain, it is muscle fatigue (that prompts subconscious movements), autonomic reflexes, and well, sleep cycles or phases also lead to the body shifting positions unconsciously.

In any case, people with CIPA are at a much higher risk of developing complications related to immobility because they do not experience the typical discomfort (pain) that triggers movement.

I can do more than that. I used to wake up sweaty half the time from sleeping as child. Now that never happens as an adult, i instead wake up with my sweater mysteriously taken off. I wish sleep self would learn to put a sweater on if it turns cold though.

Yup, putting on a sweater is a more complex task which requires deliberate planning and coordination, it requires more cognitive control as opposed to tossing your shirt which is a simple reflexive behavior. Sleepwalkers can perform highly coordinated and complex behaviors as it does not happen during REM sleep where your body is paralyzed, but sleepwalking is a distinct phenomenon, plus when you are not sleepwalking, the brain's control over motor functions is very limited.

Not everything is lost though, you can potentially learn it through repetition, you just have to do this in a half-awake state or in a state close to sleep. Do it often enough, and theoretically you may be able to perform the task during non-REM sleep.

One can make an argument we are conscious at some level when sleeping. We must be. How else would one wake up immediately when one's partner calls ones name, and sleep through cats screeming to be let out during the night :-)

I swear I used to be able to switch off my alarm clock while still fully asleep

Must be why there are apps that make you solve a math problem or take a picture of your bath room to turn off the alarm.

It's exactly what I use "alarm clock xtreme" or whatever it's called. Math puzzle, retype or even scan barcode to stop alarm. It has been invaluable, especially as you can change difficulty, I found out even half asleep I could do basic addition etc...