I'm not the world's expert in this, but I think I have a pretty good idea.
DVT is caused by some combination of immobility and hypercoagulability (increased tendency for blood clotting).
Both of these are important, so most people, who do not have a reason to by hypercoagulable, probably would not be prone to DVT in the course of normal events. This includes long airplane flights, but probably not a few weeks immobile in the ICU with a life-threatening disease.
There is a subset of people who may be prone. The most important factor is genetics, but other factors, including illness, some medications, and smoking, may play a role. (Of course, most people will not know if they are prone or not.)
Immobility is important. Obviously, sitting without moving for long periods of time will be a factor leading to development of DVTs in some people.
What is really happening with DVT formation with immobility, however, is restriction of blood flow in the legs. With normal walking and movement, blood is actively pumped in the veins; the veins are designed that way. Immobility prevents normal pumping.
Factors that restrict blood flow include:
-Dehydration: less blood volume = slower flow; also, "thicker" blood promotes clotting (makes it hypercoagulable).
-Bent legs: this actually constricts the veins in the back of the knees. If the veins are constricted, more blood can pool in the calves, thus promoting DVT. Obviously more of a problem in economy.
-Dependent legs: with the legs down, the blood has to fight more against gravity. Elevating the legs will protect against DVT. Again, seems to be more of a problem in economy.
Regarding low air pressure or air quality, there is really no evidence of this. As far as I know, there's only one very poorly designed study in humans and one tiny study in rabbits which barely meets statistical significance. Highly dubious.
So, take the first-class seat, drink lots of water, and try to walk around. I actually doubt there's much difference between a zero-degree first-class suite and your bed at home.
By the way, TED hose compress the calves and prevent blood from accumulating in the veins as well.