*sigh* This has been futile. Today I spent time cleaning the sensor using Pec Pads attached to the Sensor Swab wands along with the Eclipse solution.
First I used the Rocket Blower to make sure no dust/dirt was on the sensor then I wet cleaned it with the moistened swab per the instructions in the video. Then I used the Rocket Blower a 2nd time and wet cleaned it again. A test photo followed--still didn't get the sensor cleaned completely. So I repeated the entire thing again (twice like I did the 1st time) and took another test photo. STILL didn't get it cleaned. Grrr...
So the third time I used the Rocket Blower then really scrubbed with yet another moistened Pec Pad and Eclipse solution then used the Rocket Blower again and wet cleaned the way it should be done. Another test shot ensued. Geeze.... STILL not clean. And I used a clean Pec Pad every time (6 total).
So I took 100 continuous high speed shots (well, I don't have a grip so 'continuous' meant 5 here then 3 shots, then 2, then a few more until I reached 110--I went as fast as the D600 would allow).
Here is a 100% crop of the upper right corner of the photo. Previously it was the upper left and lower right that showed the highest amount of spotting.
So my plan is to contact Nikon Customer Service and see about getting my D600 sent out. I also want to have them check the autofocus calibration along with how well it works when holding the shutter partway to autofocus before recomposing the shot. Somehow there is an autofocus issue. At first I was having a lot of front focus issues, then I took this shot--focused on my Australian Shepherd's eyes while holding the shutter partway before recomposing and taking the shot--this is a completely unedited photo shot at f/5.6.
This definitely looks like it back focused. I've never had this issue with my D90, my N90s, or my N70 (or my previous Minolta manual focus camera). Something is off somehow. It is time to contact Nikon.