Yeah, I agree that the war would have pretty much been done in the European theater.
Whether Germany would stop fighting is a difficult question to answer. Hitler's regime was bent on world domination afterall. Frankly, I think that, with a victory on the Russian front, Hitler and his army would be free to focus their efforts on Great Britain. Conflict on the Russian front was one of the main things that kept Germany from conquering Britain in OTL. Without that conflict with Stalin's forces, Germany would have been free to focus its efforts on Britain and actually conquer it.
Now, what happens from this point forward is a bit of a mystery. Germany would want to move on to conquer America which, in my opinion, would be a more difficult job than conquering Russia.
I'd say that America, whose European allies would be reduced to underground resistance groups, would be quick to sign a peace treaty with Germany. Actual violent war with Germany would, in all likelihood, cease for America. Obviously, since Germany was an industrial power back then, a Cold War would probably ensue between it and America.
As for the Asian theater, this would all depend upon the terms of the peace treaty. Japan was one of Germany's allies but whether the peace treaty would include stipulations that America must stop its war with Japan would be a mystery. In all likelihood, I'd say this would be one of Germany's stipulations. But, in my opinion, it's a tossup.
If war on the Japanese front continued, China and America would be close allies (China already fought for the Allies in our World War II, so I don't see why it would differ in this timeline). Japan, assuming no nuclear weapons were used, would be conquered via a land invasion and would either be controlled by China (the most likely candidate) or Japan.
With a fascist Germany occupying the role that the Soviet Union had in our timeline, America's greatest fear would be for the world to fall into fascist hands. In all likelihood, America's major concern would be in Africa, particularly Egypt. I could see the Suez crisis (assuming it pops up in this timeline) being a huge foreign policy issue that could potentially explode into this timeline's Vietnam War.
Egypt (and, potentially, the rest of North Africa) would probably be split along pro- and anti-Fascist lines. Pro-Fascist Egypt would likely come to dominate and America, fearing that the other Middle Eastern countries would follow Egypt's example and become fascist (this would be this timeline's version of the Monroe Doctrine (which, in OTL, said something similar about Southeast Asia, only that pertained to stopping the spread of communism)), would enter the war along the side of anti-fascist resistance groups.
Anyway, sorry to post so much. I guess I'm going to earn the reputation on this forum for being the mega-poster.
BTW, quick question: in your timeline where Germany conquers Russia, does it conquer all of Russia or just the western part of it?