I did buy the D800 after having the oil probs with the D600.
A clear advantage of the D800 is the AF which indeed works better and faster than the D600's. Not even just when you use all AF points, but it is also considerably faster and hunts less when you just use the central AF point. Also the AF-C is better.
The 36 MP is no prob, I didn't use anything much below 1/125 even on my old analog SLRs, and at 1/125 I can shoot clean pics with the D800 as well. It is, however, true that to a small extent, an unsteady hand can become a tad more visible with the D800 than with the D600.
Regarding the lenses, if you use less than optimal lenses, they will not produce worse images on the D800 than on the D600. They just don't allow you to get the maximum out of the D800's magnificent sensor. Images taken with inferior lenses on a D800 will look the same as they look on a D600. With better lenses, the D800 just can get a bit more out of an image than the D600.
The handling of both cameras is by and large comparable, although the D800's user interface is quite different. But both are intuitive and easy to learn.
The D600 beats the D800 in terms of bulk - it is not that much smaller and lighter, but the difference is noticeable.
The 36 MP are not an issue in terms of processing, provided you have a good computer and USB3. The loading delay for D800 images in Lightroom is slightly higher than for D600 images if you use full quality (which I exclusively do), but the difference is neglectible.
My D800 has taken quite a bit of beating, it has travelled a lot in the trunk without much cushioning, and it still works flawlessly. So maybe its build is a tad rugged than that of the D600. Difficult to compare. I also won't talk much about the weather sealing - I'm less weather proof than the camera, when it rains so hard I stay inside.
One thing I can say is that the D800 copes very well with climate changes. Like out of the A/C'ed car into 45 °C, and the images still don't have any mist in them - that is nice.
And of course - no oil ;-)