Wow, Craig. Lots of good questions.
As far as I've been able to figure so far is that the nightshades have a "sort-of" vitamin D-like activity, but they screw up the calcium - phosphorus ratio in the process. This problem is compounded by a relative magnesium deficiency.
The biggest relationship that I could scrounge up between Vit. D and K and osteoarthritis was this quote from a research review abstract:
Hyperlipidemia, vitamin D, nicotine, and warfarin, alone or in various combinations, produce arterial calcification in animal models.
Since vitamin D-like activity and nicotine can be found in all the nightshades, that may be a connection. This would be further exacerbated by a relative K deficiency and a magnesium deficiency.
So, is arthritis related to a vitamin K2 deficiency? Would vitamin K2 offset some of the effects of nightshades? Do nightshades lead to heart disease through calcification of arteries?
1) Yes, it is a factor.
2) It is likely. However, it may be analagous to adding more oil to an engine that really needs the correct grade of gasoline to be used.
3) I believe that it is a major factor.
Scott made a great series of posts on his Modern Forager blog about the relationships of the fat-soluble vitamins (A,D,E,K) to each other. You might want to check those out if you haven't already.
On the last part, I have heard many people advise against consuming or heavily reducing consumption of unfermented soy and soy products. I have yet to hear any good arguments against consuming fermented soy products--this is the traditional way of preparing soy foods--this is typically the approach that WAP recommends. I think there are more benefits to natto than simply K2.