There are no ultra compact camera that meet all of your requirements.
Since you want an ultra compact that works well in low light, you are down to a choice of four cameras.
I know they don't meet all the other requirements but here are your choices:
Fujifilm F30, F31fd, F20, F40fd
They may not be considered Ultra Compact, because they are about 1.1 inch thick.
Those are the best compact cameras for low light.
There are other cameras much better in low light but they are not small cameras and they are much more expensive.
Here is a comparison of specifications:
Ok i have done a ton of research and still find myself confused with all the digicam lingo so i figured if i tell you exactly what i want in a camera someone can suggest one that has all the things in my criteria.
REQUIRED: Ultracompact, 6-7MP, Image Stabalization, Action/Sports Mode, Different Scene Modes and Color Modes (like to make pics black and white, etc), Movies with sound and preferably the ability to use zoom in the movie mode, 4x optical zoom, good in low light and indoors, and lastly GOOD IMAGE QUALITY!!
I hope you all can help as i have found all of the requirements in a Panasonic FX3 but the image quality is not that good...it looks grainy (I guess that means it has alot of noise?)
My range is $350 or less and I will be using the camera to take pictures of friends and family indoors and outdoors and some nights at the club so performing well in low light is essential as well.