Hi Kathi, and welcome to CruiseCrazies. I'm not on your cruise, but wish I were. Ours isn't too far off. Just understand, that this is your 1st cruise, but not your last.
As for your other questions, yes you can bring your own cup for the free drinks. I have seen quite a few people do it. On the oher hand, I don't really think that the cups are that small.
We were at the Blue Lagoon a couple of years ago. You take a boat ride right from the docks where your ship is moored. There may be a little walk because you have to go around the corner, but it is not a cab ride there. This is a relaxing excurrsion with some snorkelling and , as I remember, some swimming with stingrays (not dangerous). There is an area for swimming in a lagoon which your boys may enjoy.
We have never sailed with little children, so it may be hard to answer your next question, but if you feel uncomfortable dining on formal night with your kids, you can always eat at the casual buffet area. The other thing is that I think that your short red dress would be fine for the formal night. As far as women's fashion goes, you will see that there is no hard and fast rules. The clothes run the gamut.
I'm not sure which movies will be in your cabin, but I do think that they will be free. There are a few channels with different movies at different times and they do include children's movies.
I hope that this helps some. I'm sure that there are others on the board that can answer the questions about the children's program better than I can.
Enjoy the cruise. :wink:
Howard