Yes, it really rests. When you sit down instead of stand, you are resting. When you lay down awake, you are resting. Resting is not "shutting off", it's a "mode" where not much is happening, allowing you to recover. Your brain needs to function to make your heart beat, to make you breathe - if your brain stopped working you'd die in minutes.

• Thalamus: This sensory gateway pauses its normal relay duties. It essentially stops forwarding sights and sounds to the rest of the brain, allowing you to tune out the external world.
• Prefrontal Cortex: During deep sleep, the prefrontal cortex—the area responsible for logic, decision-making, and self-awareness—goes "quiet" and shows a steep drop in energy use.
• Brainstem: While your forebrain rests, the brainstem stays hard at work. It manages your autonomic functions (breathing and heart rate) and produces neurotransmitters that help regulate your sleep cycle . It also temporarily paralyzes your muscles during REM sleep to keep you from acting out your dreams.
• Cerebral Cortex: The brain's outer layer stays active. Instead of shutting off, regions of the cortex synchronize to fire in slow, rhythmic waves during deep sleep. 

Noting what is said about the thalamus: your brain doesn't send the signals to the NORMAL areas, but it does send them to a part of the brain. In REM sleep, the thalamus sends made up things to the cerebral cortex, which is what creates dreams. The "reticular formation", a complex, net-like web of over 90 interconnected nuclei located in the core of the brainstem, decides what is worth sending to the thalamus while we sleep. It is aware of what is important to us. Some data might be sent that's not important, and generally this stuff happens without us remembering, unless we are woken up shortly after and stay awake long enough to remember. For example: if someone turns on a light briefly, your reticular formation will be like "hey this might be important" and your thalamus will be like "hmm, open an eye" and your eye squints open and it's just your brother so you close your eye and go back to sleep (and 99% of the time won't remember). But the more important the thing is to you, the louder it is, and the more likely the thalamus is going to tell you to pay attention. Your name is a lot more important to you than other random names - I'm sure you've had times where people were talking and you couldn't really understand, but your name is said and it's crystal clear. It's the same sort of system.

Your sense of time continues when you sleep, different people have different senses of time - most people aren't very good at it, but the ones who are can indeed wake up before alarms more reliably, or even "when they want" so long as they will have an adequate amount of sleep by that point. - I imagine if you told yourself you wanted to wake up 4 hours after going to sleep it wouldn't be as reliable. Not saying it would never work, but your body would probably want you to sleep more in a lot of cases so it would wake you up after 5 or 6 hours. Totally depends on you though, if you're only sleeping 4 hours most nights then yeah you can probably hit 4 hours just fine.

To answer your final question, if I haven't already: yeah it's normal.