I need a new P&S - hate my coolpix!

Julia --

This year's crop of cameras might just be the one that gives you what you need. Although I will say that the need to handle "children and lots of indoor or motion shots" sets a high bar indeed.  I think sport photography and children photography tie for the most demanding photo challenges, with children being a touch harder :-) The issue is that they move a lot, and move unpredictably, so it's very hard to have your camera ready when they happen to be in the scene.

Frankly, I have tried several cameras, and only the dSLR with a seriously fast exposure and focus system are up the the challenge (and then, about 1 in five shots is any good).  Otherwise, you need to set up a photo and snap at the perfect moment, or worse, get the kids to pose (and you know the fake smiles that come from that!).  That these pictures are indoor means light is a challenge.  So pretty much anything small will be a little less than perfect, I think.

All cameras need to focus and set exposure before they can take the picture.  I bought a Canon G6 as a smaller camera than my Canon 10D dSLR, and it was supposed to be "great" at this.  But compared to the 10D (and the Canon lenses), the G6 is terrible.  That was a couple years ago, but by today's standards, the G6 is huge.  I haven't seen the G7 yet - maybe it's better.  And really all compact and point-and-shoot cameras suffer from taking a "long" time to focus and set exposure (even if that's only 1/2 second) and it gets worse in low light.  So I think this is the real challenge.

I think you're right to go for image stabilization, and I think a good zoom, which will allow you to get away from the subject.

When you are further away, you have a little more maneuverability, so can track the subject (sounds like we're hunting deer, or something, no?) and lock in for the kill... er ... shot ... er photo at the right moment. 

An important trick is that all cameras can do focus and exposure in a first phase (shutter button half-way down) and the picture as the other and much faster phase.  Even with your current camera, getting the knack of this can make a huge difference.

Red-eye happens because the camera flash is very close to the lens: eyes of people (and especially animals) are shiny, and reflect the light of the flash back at you, so if the lens and flash are an inch apart, the eyes act like a mirror shining right back.  Compact cameras and flash all cause red-eye.  "Red eye reduction" features work by sending a burst of flash at the subject's eyes which (in theory) causes the pupil to contract making the mirror of the eye smaller.  In practice, it makes taking the photo take even longer, and doesn't really work anyway.  Pro photographers who have to use flash have flash units that are separated a long way from the lens (a foot or two, or more).  One big advantage of the larger cameras is when they have a pop-up flash -- even a few extra inches can make a difference.  Another technique is to hold your finger or something up away from the camera, which draws the subjects' eyes to look there, instead of right into the lens or flash.  The best solution for red-eye is: no flash!

But zoom, again, can help on this score.  If you are away from the subject (and your camera's flash is strong enough) the subject really has to be looking right at the lens to get red-eye.  But compact cameras usually don't have very strong flashes -- cameras like the Canon S2 or S3 both have decently powerful flashes that smaller cameras may not.  Since camera makers know this, you might assume that at a camera's maximum zoom range, the flash is going to be a little weak, but if you have a 10x zoom and are only using 5x, you should be in good shape.  The best way to deal with red-eye, is to not have to use flash at all, although this may not be practical.

As to "noise", this is the presence of pixels that are the wrong color and lightness, e,g, a bright red pixel in a black background.  When you have enough of these pixels, it can make an image look lousy.  Image noise gets worse as ISO in increases, so one might think that cameras with higher maximum ISO settings are less likely to have noise at any one setting -- wouldn't you think a camera that can go to ISO 3200 would be better than one that can only go to 800?  But the proof is in the pudding, and it's really only after the cameras are tested that you find out who has the real highest ISO.  The Canon cameras tend to have low noise at high ISO.  I would be sceptical of any ISO greater than 800 in all but the best camera -- noise may be better than no picture at all, but the noise in a lower-end camera claiming to go to ISO 3200 is probably going to produce a bad result.  And the camera that will automatically set to a higher ISO (auto-ISO) as needed to deal with low light, are a good thing as long as that high ISO isn't higher than the camera can do well.  I suspect this is what people are seeing in the Panasonics you mention.

So I would wait to see what people say about the G7 -- if it is comparable to the G6, it's still a reasonably big camera, about the same size as the S3 (but without the zoom and IS).  This line is aimed more at people who wanted an SLR but couldn't afford one.  Now the SLRs are cheaper, so hopefully the G7 is a completely new design that competes with some of the better non-SLR high-end cameras.

I would have to say on the inexpensive and small end that the Panasonic TZ1 might be a good bet for you.  It's small, but it will probably be slow.  The Canon A710 IS comes from a long line of pretty good cameras so is probably a good bet -- the A700 is getting good reviews and seems like it's probably the same camera just without the IS.  The SD800 doesn't have the 6x zoom, so I would go with the A710.  The Kodak V610 also looks like a good bet.