The actual focal length at 10.7x zoom is about 66mm only.
So shooting at 1/60 or faster handheld at max. zoom should still be ok if you have reasonably steady hands.
This means no problem in most outdoor shots as it's quite bright.
As for indoor shots, if the proper exposure is 1/15 @ISO 200, then you could pump up the ISO to 800 and shoot at 1/60. The advantage is that you can avoid too much motion blur when shooting at 1/60 vs 1/15.
If avoiding motion blur is paramount, then you would need at least 1/125 or faster. In such cases where shutter speed is the priority, having good low noise performance is better than having IS because the former may enable you to shoot with a shutter speed above 1/125 while the latter does nothing to the shutter speed. For example, if an indoor shot at max. zoom (66mm) requires 1/15 @ISO 100, then you could shoot at 1/15@ISO 100 if you have IS or shoot at 1/125 @ISO800 if you have good noise performance. The 1/15@ISO 100 shot with IS has better details but 1/15 may have motion blur which you do want.
If you're shooting at the short side of the zoom (the actual focal length is only slightly more than 6mm when it's 28mm equivalent on 35mmm format), the IS is not required in most situations except very dark conditions (which may be better just use the flash instead if there is no restriction).
To me, S6500fd is a better choice than cameras like Canon S3 IS which has terrible noise performance at ISO 200 and 400.
Don't think so. Longest focal length IS 66mm (or thereabouts) if your sensor has the dimensions of a 35mm film. But it uses a teensy-weensy sensor (relatively speaking) so you have still to use the 35mm equivalent, which is 300mm. The smaller sensor diameter amplifies shakes and wobbles, corresponding to the equivalent length. So you still have to use 1/300s shutter speed, or whichever is closest. Same holds true for the shorter focal lengths.