The opposite is true, Android sleeps unless something explicitly keeps it awake - i.o.w. a process holds a wake lock. Where PC operating systems go to sleep when there is nothing to do for a while, Android sleeps until being woken by user interaction or scheduled timer events. It then performs the task(s) after which it goes to sleep again. A music player on a PC would keep the machine awake, a similarly built player would stop playing after a few seconds on Android unless the developer made sure to schedule wakeups or used the big hammer of requesting a wake lock. Spurious wake locks used to be a common cause of drained batteries until Android started to put limits on background processes.

This is false. Android is famously narcoleptic, if you don't have a kernel wake lock, you sleep. Doze mode is orthogonal to CPU sleep.

This usef to be the case perhaps. Check out documentation on SystemSuspend and doze mode in current Android.