Based on the side by side you posted, I see the Panasonic Lumix DMC-FZ200 as having a few big advantages over the others, as well as a couple of disadvantages.
First the disadvantages -- the FZ200 is much bigger than the other cameras. It is not a compact camera. So if you truly want a compact camera that can fit in your pocket or easily fit in your purse, then cross if off the list.
Also, it's about $550 from what I saw online -- For just $100 more, you could get the RX100. Or for around that price, there are lots of other options. You can get the Sony Nex5N for about $500, and it is a much better all-around camera than the FZ200. You can even get a true SLR for around that price.
Now the advantages of the FZ200 compared to the other 3 on your list, at least according to the comparison you posted:
-- Faster lens. Aperture of 2.8, and it appears to be that way throughout the zoom. That's a much faster lens than the other 3 cameras, especially at maximum zoom.
-- Ability to shoot in RAW mode. This is great if you like to do post-processing in lightroom, or want to build your own HDR images.
-- More manual control than the others. More manual control means more freedom to override the auto and get the picture you really want.
But for the size and price of the FZ200, there are other cameras that should be on your list. Many of those cameras may have inferior zoom, but will be superior in every other way, including in image quality.
That said, for the specific purpose you've been doing -- Shooting your daughter from the stands, the FZ200 is a great choice. The extended zoom and the 2.8 aperture would really do a great job in that situation.
If you were to add other considerations in that price range:
You could get a Canon Rebel T2 with 2 lenses, including a 75-300mm zoom for just over $600:
http://www.amazon.com/Canon-Digital...=sr_1_4?s=photo&ie=UTF8&qid=1349876598&sr=1-4
For under $500, the Sony Cyber-shot DSC-HX200V .
For $500, the Sony Nex-5N:
http://store.sony.com/p/Sony-NEX-5-18-55mm-lens/en/p/NEX5NK/B. It lacks the super zoom, without adding a lens. But you would have a SLR-sized sensor in a pretty small body. And can add a better zoom later on. (A good zoom lens can be added for $350).
Also in that price range, you can get some great refurbished and used cameras. Try B&H Photo, for example:
A Nikon D3100 with 2 lenses (including a zoom lens) for $515
A Sony Alpha A55 with kit lens for $530. (Since it takes old Minolta lenses, you can actually add a zoom lens cheaply, under $100. For my Sony SLR, I basically just use old Minolta lenses.)
So really, if you are looking in the $500+ range, and are looking beyond compacts, then you have lots of great options beyond the FX200.
Of the 3 compacts that you listed, the Sony Cybershot HX30V is probably the best but they are all pretty close. I happen to think Sony excels at smaller-sized cameras. (The NEX series seems better than rival bridge cameras. The RX100 is the best pure compact.). Of the 3 compacts you listed, the HX30V has superior ISO range, more megapixels, and a slightly faster lens than the other 2.
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