AllAmerican's answer above is really good, as are a lot of others on this thread. Chiming in (relatively) fresh off deployment here. Keep in mind all of my experience is CRU/DES. I don't know what the gators do other than show up for photo exercises on occasion.
Generally speaking sleep is at a premium, but there is a great degree of inequality in its distribution. Engineers tend to get the least, then Combat Systems, Operations, and Weapons in that order - at least among first/second tour DIVOs. Generally speaking the more successful you are, the less sleep you'll get as you're trusted with more on more responsibilities. At some point you get used to making due with less - though how much less varies person by person. There are guys I sailed with that could make due for long periods on 3-4 hours, with a cat nap here or there. Generally speaking I found if I consistently got less than five I was a real jerk to be around.
Sleep can also be a feast/famine thing. My first month onboard, during my second tour, we experienced one major casualty in a system I owned after another, seemingly every three or four hours. I almost never got more than four hours of sleep a night. That entire patrol three month patrol was an absolute shit show and by the end I was worn absolutely ragged between requalifying OOD, qualifying a combat watch, learning my divisional stuff, and getting up to speed on theater publications.
The following patrol just a month later was the easiest three months of my career, in which I was only woken at night with casualties handful of times and we were on four section watch (six on, 18 off). I even had time to catch up on some personal reading.
The deployment after that I qualified another watch station, we were short some people, and I spent the entire nine months either standing the 2-7/15-18 or six on six off. I typically got about five hours between 1900-0100 and snagged an hour or two in the morning after Khaki call whenever I got a chance before the morning /afternoon meeting rotation started. When I was dead tired (six/six, if we had evening events or casualties) I'd pretty frequently meals.
People show up late - it happens. You'll get woken up. If it's a one off people tend to be pretty chill. If it's a pattern... they'll find ways to correct it. Set multiple alarms if you have to. Don't make it a pattern. In time though, as others said, you'll get accustomed to the noises of the ship. When I was on the DDG in overflow the engineering divos could sleep through Neanderthal divos shuffling around in the narrow passage and slamming their racks, but would wake up anytime we had an engine casualty before it could be called away. On the cruiser despite the ungodly racket of flight quarters I was never woken up by it (or the 1MC) except when it passed words meaningful to me or the phone rang (and both of these woke me like the dead).
Pretty amazing what you could adapt to.
Protection of sleep is getting marginally better over time - the O5 commands are better at this than the O6. There's still an expectation that you soldier on through rough days, but generally speaking on the destroyer I never got intentionally set up by the rotation to get screwed. With the O6s... old habits die hard.
