First your arithmetic is wrong. You have 703 images and have used 37% (100-63=37) of you battery. That extrapolates to 1900.
That is far short of 4000, but better than you calculated. I've never used a D3 or D3s, so I cant say from personal experience what to expect. I have found on other Nikon DSLRs that setting up the camera with the menu seems to use a lot of power. Of course a lot of image review and playback also uses up power, maybe in your case that's the real culprit.
I would be suspicious of a battery age of 1 as well. I had a D300 for 5 years and taken about 40K photos with it. The battery still shows full life. Possibly using the calibration ability of the charger could restore some life or change the apparent capacity. I had a new battery for my D4 that showed a life at 1 and it seemed to go down quickly, calibrating it reset the life and gave me more indicated exposure life.
Note I've shot 5700 shots on one D4 battery with very little chimping, I'd expect similar capacity on a D3s.
I can't answer your other questions about shutter life, but I suspect that D3/D3s shutter cycle rating is much higher than the D200.
Gary in SE Michigan, USA. Nikonians membership - My most important photographic investment, after the camera. D4, D810, D300 (720nm IR conversion), D90, F6, FM3a (black), FM2n (chrome) YashicaMat 124, Graflex Speed Graphic 4x5 My Nikonians Gallery & Our Chapter Gallery