There was a whole series on seasickness recently, with lots of helpful info. Go to "General Cruise Chat" and click on that main subject title - it should bring up a list of topics recently posted there, and the seasickness one should be near the top.
You didn't say where you are going, so answers to your weather question may be more difficult to get. Never any guarantees - probably hot with variable clouds, assuming you're cruising the Caribbean. Be prepared for rain or sun, and remember the sun in the Caribbean is quite strong, so sunscreen is essential.
Your kids should have a great time - Royal Caribbean has lots to offer those age groups. Have a great time!
Just sailed in mid July 03. I didn't bring anything on board thinking I'd be fine. When I got there, the movement did bother me only a couple days. And to my surprise, they offered FREE Bonine pills at the Guest Relations Desk. So no worries, just grab some on board if you're feeling it. They did make me a little drowsy when I took them, so maybe take only half of the pill at a time.
Also, there were great programs for both kids and teens. They seemed to be having a great time.
Weather in August should be fine. Humid and hot, maybe scattered showers, but nothing that doesn't feel refreshing. Possible hurricanes, but the boat will redirect the course if in direct path.