Goering inherits a Germany that is at its greatest extent and includes all Germans except for Danzig. He pretty much has everything he would want in Europe. His foreign policy aims were much more in the Wilhelmine Imperialist mode than Hitler's belief in Lebensraum.
I think the rest of Czechoslovakia is likely to be left alone. He may begin negotiations with Poland over Danzig and a secure land access route to East Prussia. IOTL, Ribbentropp asked for this in October 1938 in exchange for an extension of the German-Polish Non-Aggression. That was turned down so soon after the seizure of the Sudetenland. But with Hitler gone and Goering refraining from other actions, some sort of agreement is theoretically possible.
If Goering can achieve that, he'll boost his popularity with a substantial foreign policy victory of his own. Then I think he'll move to the two areas he probably wants most - 1) increased respectability with the Western powers, and 2) recovery of former German colonies. After consolidating power at home, I could see Goering going on some kind of "charm offensive" in France and Britain to reassure them of his peaceful intentions and gain support for the return of some colonies. It might work, although German colonial gains would be limited and paid for by some kind of concession.
I don't see Goering starting any major war unless major German interests were at risk, or if the war would be limited in nature. So I see World War II being avoided. Presumably, Goering is able to maintain some kind of anti-Bolshevik consensus in Europe and contain the Soviet Union.
He would likely offer his services to mediate the Sino-Japanese War, although it's likely the Japanese have dug themsleves too big a hole that they can accept any reasonable peace and still save face. If it ever happened though, he'd have huge prestige.
With no war in Europe in 1940, FDR does not run for a third term nor would Wilkie get the GOP nomination. We have an entirely different 1940 election. Maybe James Farley versus Robert Taft?
There'd also be a UK general election in 1940. I think the Conservatives would have won since they secured "peace in our time" although with Chamberlain dying of cancer, someone else would be PM. It won't be Churchill. It can't be Halifax. Perhaps Eden? Who would be the most likely candidate.
I think Europe will shuffle along without too much incident. There might be concerns of the USSR, and there might be a growing "European Position" in regards to the Sino-Japanese War, but I don't see either turning into a war.