I'd recommend you run some tests for yourself and determine what apertures will work for you. Often with telephotos, you're isolating the subject and therefore things like corner softness and light falloff, which is what you improve by stopping down, don't matter much anyway.
I had a telephoto lens which I always used stopped down since that's what the prevailing wisdom said was necessary. Then, when I was selling the lens, a potential customer asked me for an example shot wide-open. I did the shot practicing my most stringent shot discipline (weighted tripod and lens, live view focus, mirror-up delay...) and found that wide open was actually the sharpest aperture!
In DX mode, your 16-85 is the same lens with the same performance characteristics regardless of D600 or D80. I'd expect better results on D600, particularly at higher ISOs, mainly due to a few generations newer sensor tech. To take full advantage of your D600 however, you'll want an FX mid-range zoom eventually.