I often take two photos of my cats, if or when they hold still, this way I can deal with the "red eye effect" later in editing.
I get rid of that flash and crazy eyed look by transferring the non-flash eyes to the flash version which often has the best color and focus. My (point and shoot) Fuji FinePix F20 does a good job and has a setting to take both (flash and non-flash) each time. I usually keep it on this setting because it also works for reflections in glass, etc. and means I automatically have a "fill flash" outdoors.
Here is an example of one of my kitties.
