Well, if you have a "stateroom" the same size as ours, 15-20 suitcases might be a bit difficult. There are no limits to the amount of luggage you can bring on except the size of your room. We had the smallest room because we booked about 4 months before we sailed. The storage consisted of a built in wardrobe, a small built in dresser, a built in desk, and I think what you could stow under the bed. Between the two of us, we had 5 cases (large) and carry on. I shipped the majority of the stuff I was sending over and could live without for a while.
Honestly, at a push I would say if you're travelling with someone, 8 would be the limit. Keep in mind, you will have to be able to manoeuvre your stuff through customs in South Hampton, which is extremely disorganized compared to embarking in Brooklyn (and I heard the same is true going the other way). I would really suggest looking at some cheap shipping options for some of it. In the bigger staterooms there might be more room for stuff, but trust me, we were pushed to the limit disembarking with what we had, and we didn't have any animals with us. They do transport your luggage to a place to pick it up in the port, but it still is a bit crazy trying to get through the masses of people.
There is at least one indoor pool, but I never saw anyone in it. There are films, a planetarium, RADA productions, a library, a pub (with karaoke, quizes and so on), lectures, a cyber cafe (although we never used it), a game room/hallway thing, a casino, a night club, a wine bar... It sounds a bit stodgeyish, but we had a lot of fun. Make sure you get tickets for things like the planetarium as soon as they are available or wait until later in the trip to see them. There aren't a tonne of films shown, but IIRC, they played two different ones in the evening and a classic one that was a subject of one of the lectures given. You can also bring films on your laptop, but really, we always found stuff to do, and I was a lot less bored than I thought I would be. And we had a huge storm at sea, so we couldn't do what I envisioned the crossing to be like: days spent on a lounger under a blanket with a book.