If it were biologically possible, other organisms would have evolved that capability. There’s some fundamental, biological reason why all animals sleep.
Again, we don’t know that. It could easily be that the adaptations needed to operate well at night (in addition to during the day) just aren’t worth the energetic cost or they entail a large morphological compromise. The thing about evolution is it doesn’t give you its reasons.
Exactly, evolution also will leave things at "good enough for function". It may well be that our sleep cycle so happened to level off at "good enough" considering our other evolutionary constraints.
Definitely an area of study that seems interesting to me.