I don't happen to have a set of scales to hand, but it's reassuringly chunky, rather than outright heavy. It's heavy enough for your hands to shake a little when you're holding it with the zoom up high, but I'd imagine most 10x zoom cameras would do the same.
As for the S6500, I don't honestly know. It seems to have a higher resolution, as well as better manual zoom and focus controls, but I'm not sure if it's worth extra over an S5600. I think I'd probably just go for the S6500 if I were in your position, but I'm the kind of person who goes for the best I can afford. Whether you'll use the extra features of the S6500, I don't know.
Why not take a wander down to any local camera shops and have a play, see if the S6500 is to your liking?