"You should sleep more."
Translation:Ты должен больше спать.
Right, большевики́ is related to бо́льше. The original party, РСДРП, got split into two fractions. More (бо́льше) members of the party belonged to what became большевики́, while the fraction that had less (ме́ньше) people became known as меньшевики́.
Yes, «Ты должна́ бо́льше спа́ть» should be used when addressing women.
Your sentence sounds unnatural.
Намного 'much' cannot modify спать directly. You need to combine it with some other adverb, such as «Вам нужно спать намного больше» 'you need to sleep much more'. But then it doesn’t correspond to the English sentence.