I'll agree with the general consensus...hopefully you're just looking for the best combination of these abilities for $250 or less, and not actually expecting the camera to do all of those things 100%. Unfortunately there are always compromises with cameras (as much as any other product or device)...you can't have it all. A P&S camera will not be able to perform all that well in some of the situations you mentioned...most cameras won't. If and when you find one that will, it likely won't be under $250, and it won't be small and portable. Just the nature of the game! Sort of like wanting a car that can get 40MPG, fit 8 people comfortably, tow 10,000 Lbs, go off road, go 170MPH, and cost less than $15,000. You're going to have to throw away a few of those dreams!
And then there's the other factor. The camera isn't a miracle device...so without good photography skills, there are some things the camera isn't doing - the photographer is. Getting good shots with sports requires not only a fast, capable camera, but a good, skilled photographer. Many times when you see that 'perfect shot' of a sports moment, it involved a photographer with amazing timing, a good eye, excellent skill with a camera, and they probably have another 100+ shots taken around the same time that are not so perfect. Either that, or they are the photographic equivalent of winning the lottery - just an astoundingly lucky person who captured that precise moment with little or no skill!
The only other thing I'd throw in is that you may be limiting yourself unreasonably by only looking for cameras that take SD cards. I honestly can't see any reason someone would choose $250 worth of electronics based on whether or not it is compatable with a $20 memory card. Memory is so cheap nowadays that having a particular type of card is a fairly poor reason to choose a particular camera. Sort of like redesigning your whole kitchen to match your toaster.
Whether it be SD, SDHC, Memorystick, MS Pro Duo, XD, or any other card type, you can buy 4GB worth of memory for usually $25 and under. So what you really should do is look for the best camera, with the best features, that is the most comfortable for you, and the design you like the most, at the best price, and get that one. Then, go spend $20 on whatever memory card type it takes. Why limit yourself on brands to select from over such an inexpensive item?
As for batteries...you should be fine there. I don't really know of any modern cameras that rely on a USB charge, and do not have interchangeable batteries. All cameras today will use either rechargeable AA batteries, or lithium proprietary batteries. Both have advantages and disadvantages - Rechargeable AAs can be bought cheaply, changed easily and often, and can use regular AAs in a pinch if needed, while Lithium proprietary batteries often have longer shooting times, can store much longer and keep charge, and often allow the camera to give you actual minutes-remaining time readouts. Both would charge by plugging into a standard wall outlet.
I'd recommend shopping features - not brands. Look for the zoom range you want, the LCD screen size you want, the controls or specific features you want, and most imporantly, a design that is the right size and comfortable to hold and shoot with for you...then shop and compare prices to narrow the choice to the right one for you. You'll find very good cameras today by Casio, Canon, Fuji, Kodak, Nikon, Panasonic, Pentax, Olympus, Sony, Ricoh, and even a few smaller brands have a good one here or there. Stick with the larger name brands above that you've heard of, and you'll likely be fine. A decent photographer would be able to take fine photos with any of those brands above - the name on the front really doesn't have too much of an impact!
