The Canon SX1 can do everything you're asking, but it cannot be hooked to the PS3(that I know of). What it can do is be connected to the TV by the HDMI of the camera and watched directly.
I am looking for a compact camera with 10 megapixels or more and the capability to shoot HD video, both 1080 and 720. Also, I would like to hook it up to my PS3 (HDMI)to watch on my HD home theater. Is there anyone out there that can guide me to the small selection this search would bring up? Thank you.