Hi, I have just returned from a caribbean cruise with Royal Caribbean and had a fab holiday. There were quite a number of young children and teenagers on board. We walked around the ship one night and noticed a kiddies club for young children, children looked to me to be having a ball. Went to top deck and shocked at amount of teenagers as obviously well catered for as we saw very little of them on the ship. Shows also family orientated. Food caters for all ages, fabulous. Play facilities at pools very good for children and they can pull their own ice cream all day!! We were on Freedom of Seas. Staff love the children.
Forget to give you full answer, no, you don't have to dress up. As it was our first cruise, we hired the tuxedo, but really a pair of trousers and shirt are formal enough. We booked for the second sitting in the dining room as this was in our overall package, we only ate there twice and one night was the formal night. Windjammer was our personal choice, huge choice of food for adults and children, but just to let you know if children peckish before bed, there is pizza up to very late and a 24 hour cafe serving lovely rolls, cakes, tea, coffee etc..
Re dining: it depends on the ship - currently MyTime dining is on 7 or 8 ships, otherwise it is 2 dining times of 6.15 and 8.30 approx in the main dining room. There is always the buffet where you can eat when you want. And depending on the ship there are other dining venues too. Yes there are 2 formal nights per 7n cruise - you will look out of place if you do not dress accordingly.
The children's clubs start from age 3, so the 2 year old would not be eligible. Really, you would do well to study the Royal Caribbean website, where there is a lot of information. You might also want to look at P&O's Ventura, which seems aimed at families with small children.