The pools will be open and on our NCL trip it was heated. Do you have the ship picked yet? Some ships have covered pools but I don't know if NCL has any with covered pools in Alaska.
There's plenty to do for kids on the ship. If they like any sort of history they'll like Alaska even more. There are programs for kids of all ages. The 16 yr old won't have to hang with 10 yr olds.
We've been 3 times and every trip the weather changed a lot. Sunny and clear to rainy, on the same day. Dress in layers and you'll be fine.
Budget has a lot to do with what you do at the ports. Anything from expensive flight-seeing to just walking and shopping the towns.
There's always something to do on the ship for all ages. I'm always really surprised when I hear people say they were bored on an Alaskan cruise. The scenery from the ship alone is unbelievable.
If you haven't booked yet, I do suggest seeing Glacier Bay. Spectacular!