Congratulations on finding this site!!! I have personally found it to be quite helpful. I will try to adress all of your questions, so this may be quite lengthy.
1. In my experience, one of the biggest expenses on the ship are alcoholic beverages. My wife and I were on a 7 night on the Adventure of the Seas out of San Juan a few years ago and our bar bill was $800. If you don't drink much, that would save you a significant amount of money. Also, if you book your shore excursions ahead of time and pay for them before you leave, you won't have that expense while on the ship. I think a good estimate is up to $100 per person per day, but that is probably on the high side and you could easily spend less than $50 per day. Another important concept is the fact that on the ship, they do not take cash, except for extra tips (e.g. room service) or in the casino. Everywhere else you use your sea pass card and it goes on your sea pass account. Typically, this account is tied to your credit card which will be billed the entire amount at the end of the sailing. Some people choose to deposit cash in their account at the beginning of the cruise and pay that way. Now when it comes to spending money in the various ports of call, that is different. You will use cash or credit cards for those purchases. It costs money for taxi cabs, admission to beaches, shopping, etc. Again, how much each of you spends will be up to you. I personally think you can have a great time without spending much money, however, I tend to spend more than average. During the above mentioned cruise, I spent $1500 cash while in port shopping and had a sea pass account bill of $1350 for a grand total of $2850. I did not skimp on anything and we did whatever we wanted to.
2. There is not a dinner theatre on board the Royal Caribbean ships. There are nightly shows scheduled around both the early and late dinner seatings so any passenger can enjoy dinner in the dining room and also then attend the show. There are certain dress codes applicable to both the dining room and the buffet that can be found on Royal Caribbean's web-site (I'm sure some internet saavy person will submit a link for this). Basically, the dress code for the dining room is sort of business casual with no shorts or tank tops allowed. There will be at least two formal nights on the cruise which are supposed to be more stringent, but my experience has been that they are not strictly enforced. As far as the shows, I do not think there is a specific dress code in the theater and you will see people wearing all types of clothing. The bottom line is that every day and night you will see people dressed up and down and you should feel comfortable in most things you might wear. Just don't show up in the dining room in jeans and a T-shirt or in swimwear and you should be fine, although you may feel underdressed on the formal nights if you don't do something a little more special.
The bottom line is that you should be prepared to spend a fair amount of money during the cruise if you want to have extra fun. The cruise is designed so that if you wanted to only pay the cruise fare and just walk around while in port, you could eat as much food as you want and not spend one more penny (except for gratuities) out of your pocket. My advice is to check your sea pass account frequently so your bill doesn't surprise you. You will receive a paper copy during the last night on board and can check it anytime on the TV set or at the pursers desk.
Please feel free to ask any more specific questions as I know my answer may be a bit confusing.