Think about the impact on the whole society; anyone you meet anywhere could be a virtual personality.

That doesn't seem to imply much. It's still some distinct personality. What should have an impact is the fact that now there are two personalities inhabiting a single body at different times: when you meet me at daytime, it's really me, but when you meet me at night - that's a different person. Unless I've borked my "sleep" schedule and that's still me; then I might be not-me at some time during the day. That should... take some getting used to.

Also, doesn't the body need sleep, only a (part of the) brain?

What does it think it do during 16 hours of your uptime?

False memories. Your choice. You have an equivalent of full hypnotic power over them. To avoid too much work with programming them every day, a reasonable default choice would be to make them remember everything but think that they did it.

I see. It doesn't make sense to make those memories too false, though, or the reality will take increasingly more effort to cover up. Suppose, I decide to start going to gym and conceal it from my alter-ego. Suddenly they will notice that their body started to bulk up for no apparent reason.