The first need is going to be the hardest. While your eyes read the light in a gym as fairly bright, to a camera, the light is quite dark. What this means is that to get an adequate exposure, one of three things has to happen: the ISO (the camera's sensitivity setting) needs to be higher, the lens aperture needs to be wider, or the shutter speed needs to be slower. You do not want a slower shutter speed for sports, because your boys will end up blurry. Most camera lenses have a narrower aperture when zoomed in, so you end up needing to adjust the ISO higher. The FZ200 is an exception, because its lens aperture (that the f/2.8 number in the specs; the others drop to f/5.6) can be wide through all focal lengths. Put another way, If you take a photo with the FZ200 at ISO 800, aperture f/2.8, and shutter speed 1/500, the other cameras have to lower the shutter speed to 1/125, or raise the ISO to 3200 (or split the difference with a shutter speed of 1/250 and ISO 1600)...which will either cause the photos to be blurry and/or noisier.
So of the three cameras, the Panasonic has the advantage. Note that these superzoom cameras use the same size sensor as much cheaper "point-and-shoot" cameras; photos taken at a high ISO level will look fine when viewed to fit a laptop screen, but rather bad when viewed in their native size.
I am looking to buy a camera for the following needs: action shots of my boys playing sports (need adequate Zoom to take pictures from bleachers and not look like ants, no blur when in motion, and being able to take consecutive shots quickly); family photos for birthday parties, family gatherings, school events, etc (in both outdoor and indoor light); ability to take video of all of the above as well. Capturing family memories is my main interest. I know very little about photography, so I need it to be fairly simple. Have read about Panasonic FZ200, Nikon Coolpix P520, Canon SX-50HS. Not sure which of these, or any others, will best meet my needs. Any recommendations?