Does "mess" mean a common sleeping area?
In short, yes. Although it's descended from the RN, and thus sleeping spaces, recreational spaces and eating spaces are all called messes on RAN vessels, because very early sailing vessels used the same space for all three purposes, cafe has taken over from mess for the eating space in the last 20 odd years.
Army and Air Force, on the other hand, call only the eating spaces messes, leads to some confusion whenever they arrive on board.
To add fuel to the mix, on a shore base our eating spaces are called the mess, recreation spaces are called breakout rooms, common areas or bars. Blocks of cabins are called accommodation blocks and cabins, I suppose it can be a little confusing to the uninitiated, but you get used to it. If there's one thing the RAN isn't, it's consistent!
The Burkes have almost twice the crew the crew of the Brisbane for not much larger.
European designs always put more room into the crew spaces, their design philosophy calls for more comfortable crew quarters. RAN units also tend to be minimum manned, so we end up with a relatively small crew for a ship that size. Many long years ago now, I was on a 2550 ton River Class DE that had a crew of 270, now that was tight quarters! Although the 3370 ton Perth Class DDG had it beat, with a crew of 330.
So yeah, crew sizes are a bit relative. :)