In a world where humans have 'evolved' to the point where their bodies are mostly electronics, save for the brain and spine, would they still need the standard seven hours of sleep to function normally? If not, how much would the human brain manage to rely on?
I feel like you're searching for a way to lengthen the time they can be awake as much as possible. Humans absolutely need sleep. Every animal in existence does. In a study with a creature deep in the dark, that can't possibly have a day-night cycle, they found that even that creature has certain pauzes that could be sleep in it's eating schedule, even though it would be benificial to keep eating. But we don't understand why. I talked once to a psychologist with over 40 years of research into sleeping, who said: "all I can tell you is that we need to sleep because we get sleepy".
What we do see is that many regenerative processes take place in this time. Healing goes faster and better, nerves are more resilient, memory is is stored, pathways are strengthened or removed. During sleep we see a lot of good things and it's likely that even in the best scenario you still need sleep for these processes to occur. There are techniques to reduce your sleep to even 4 hours a day by spreading it out over the day, but this isn't healthy and often requires a period of long sleep to compensate.
To fix this, we can check creatures more evolved than us in this field. Dolfins sleep one brain half at a time. They are less active, but still very much ready for action if required. For cyborgs it would probably mean a period of less decision making, but they are still active during. Make most pressing decisions beforehand and then just make small changes during the sleeping time if a brain half. That way you can have little to no downtime and the full brain can be roused quickly if necessary.
Many of the humans sleep all of the time.
The AIs that govern the body feed the humans excellent dreams. The AIs keep the bodies fit and healthy. Should the human occupant wish to wake and use his or her actual body he is welcome to do so. But the AIs use them the rest of the time.
In actuality there are times when it is inconvenient for the AI to have the human wake up. The AI has work to do. It is often easier to keep the human asleep...
Strictly speaking yes, but that does not mean there are not work-arounds
While your brain will still need to sleep as much as it does now, your body is now a machine which means you can program yourself to do menial tasks while you sleep. Washing laundry and dishes, mowing your lawn, sweeping, mopping, basically all those mindless things you have to do throughout the day, but don't like doing could be turned into a sort of highly complex sleep walking behavior.
How it could work is you set your body to learning mode whenever you are doing menial tasks to train its onboard AI to do things, then before you go to bed, you queue up behaviors like they are macro scripts to run after you are asleep. If all of your sensory organs are mechanical, your body can turn off external input to your brain while it does things to make sure it does not accidentally wake you up (unless it needs to consult you or warn you about something).
Then come morning, you wake up fully refreshed to a nice clean home... or whatever other benefit you would otherwise get from having stayed active during that time frame.
This is really only one small advantage to having an AI integrated cyborg, but there are so many others that it feels like an oversight not to include one. Humans are really good at things AIs are bad at, and vice versa; so, a future where every Human has an integrated AI (or vise versa depending on your perspective) could be an interested approach to allowing humanity to cope with a technological singularity. Need to solve a complex math problem or remember some detail that you may need to perfectly recall a few years from now, use your AI "brain". Need to solve a problem that you've never encountered before, let the human brain do it.
As far as I’m aware we don’t even know why all “higher” animals need sleep.
If it’s just to get rid of “waste chemicals” which have accumulated while being awake it could probably be sped up with technology. If it’s an inherent “processing downtime” required by neurons there is no way around it.
Animals require anywhere between 2 hours (horses) and 20 hours (bats) of sleep. At the least you could hand-wave some drugs which make your cyborgs fall asleep quicker and sleep more deeply, therefore requiring less sleep overall.
If the brain is still 100% human, then yes you would still require the 7+ hours of sleep to be healthy. This is because anything less can lead to fatigue symptoms (like memory loss/concentration issues) and even the development of mental disorders like depression. The brain often seems more dependent on sleep then the the rest of our body.