Not generally but I do have a king betta who is only active at night or when it`s very dark. Other than that he rests when it`s light but my other bettas are active then so it may just be him!
No, they tend to be awake all through the day. I never see any of my fish sleep.
I've seen my bettas sleep both during daylight hours and at night, and I've seen them active during those times too, so... they're whatever they feel like being, I guess.