I am not sure that any digital camera will totally meet your #2 and #5 requirements.
Here is what is available:
Canon - S2-IS and S3-IS
Kodak - Z612 and P712
Panasonic - FZ30 and FZ7
Sony - H2 and H5
All of the cameras are under $500.
Note: The Panasonic cameras low light performance is not as good as the other cameras.
Here are the specifications:
I did not include any DSLR (digital single lens reflex) cameras, because the cost would be into the thousands of dollars for camera and necessary lenses.
I'm planning to buy a new digital camera with following basic features
1. 10x (or above) optical zoom
2. Need to get stable image even if the object is moving or shaky
3. need to have 5-8 MP
4. A 2 inch (or more) LCD
5. Able to take pictures in bright sun as well dim light or in night
Please do send me the camera models whichever you think is the best suiting the above criteria. Also cost is not a factor for me.