I'm not sure what time the casino opens, but I thought I'd pass along our experience with an under the casino cabin. It was fine, even in the early mornings and late at night. I'd pick being under the casino long before I'd want to be under the galley. I was sure that we'd hear the bells ringing all night but other than a few minor bouts of noise, we slept like a rock, kids included, it was a pleasant surprise.
When you're out to sea, it opens fairly early. (I'd guess 8 A.M. or so) but there's relatively little action there. Of course, in port, it doesn't open at all, so there's no problem. I don't think I ever had a cabin directly over, or under the casino, but I too would guess that the noise won't be a problem. Have a GREAT cruise.
Just looked at an old Capers from the "Miracle". Sea day, and the slots opened at 9 A.M.
Generally 8am or 9am for slots and noon for the rest of the place only on sea days. As mentioned its closed in port days. Not to worry...
2008-Valor and Glory
2009-Spirit and Splendor
2010-Freedom of the Seas and Mariner of the Seas
2012-Allure of the Seas and Navigator of the Seas