The number of formal nights depends mainly on the number of days cruising. If it is longer than 9/10 days and not more than 14 then there will likely be 3 formal nights. The first is usually after the first day at sea, one other is the 2nd last night on the ship with the third in the middle somewhere, but you might not find that one out until you join your ship. Not sure with RC that you can book on-line before you join the ship you may have to be quick and get in there once you know for certain, but others might add further to this advice.
On Monday booked online from the Royal Caribbean website two of the alternative restaurants on Brilliance of the Seas. However as regards dress I believe the "pay extra" alternative restaurants request the same dress code as the main restaurant. This has been my experience on other cruise lines. It may be different on Royal Caribbean so it would be a good idea to check this before booking.