I think you need to know some background information concerning the division of Germany.
There was one Germany, the Deutsches Reich. The Allies defeated it and divided it into four parts: the British, French and Americans in the West and the Soviets in the East.
It was soon apparent that after the party more and more rifts were opening. What to do with the Germans? While the Allies all said during the war that they would work together, it was soon clear that this would not work. The Soviets wanted reparations and moved heavy industry out of East Germany. Much more important for the West was the instantiation of a one-party system and centrally governed economy like the other Eastern Bloc countries.
The Western Allies on the other hand weren't so keen to have a socialist regime. While revenge plans like Morgenthau (dismantle Germany into an agrarian state) sounded good on the paper, they opened too many questions. Who should pay for the continued occupation of Germany? In a land which doesn't have much natural resources, would you destroy the capability of a country to sustain itself ? And if the Soviet Union sooner or later recovers, you have not only lost a possible ally, but exposed your borders. So the Marshall plan (with Soviet protests) started and West Germany got a new economy and a political system modelled by the West.
So during the Cold War it was apparent that the path was divided. The three Western Allies created FRG, the Soviets shortly the GDR.
And for both people there was only one Germany!
In the west you were told the evil Soviets and their supporters are illegally occupying the Eastern part of Germany. The "GDR" was never considered to be its own country, even a long time after its declaration it was only called SBZ (Soviet occupied zone in German) until 1972. So every GDR citizen was considered as a German and eligible for immigration.
In the east you were told the evil fascists have conspired to work with the capitalists to create a puppet state under their control. The western allies broke their promise to work together and only if the rest of Germany would understand the fairness of the socialist system they would rise and join their brothers in the East. If you defected into the East, you were welcomed.
(These were offical standings. The Germans on both sides did not care much about the propaganda, but both knew that East Germany was repressive, to say the least. The eastern version was not entirely without merit: many Nazis escaped, survived denazification unscathed or worked for the Americans. And as a communist you would have been in trouble in the FRG during the early years.)
So on both sides, there was only one Germany, which was unfortunately partially under control of their foes. After more and more Germans defected from the East to the West and the wish for reunification in the East was suppressed by violence during the 17th July, the Iron Curtain was built.
But what was your legal status as far as East Germany and her allies were concerned?
You were a traitor, criminal and guilty of Republikflucht (2–8 years prison under bad conditions). There are known assassination attempts in West Germany, like Wolfgang Welsch and Fritz Eigendorf.
Some people who were captured were offered for ransom because the GDR was always interested in western currency.
Could you travel to the Eastern bloc countries immediately on your freshly minted West German passport?
Snort. All eastern bloc countries had a good intelligence and, with exception of e.g. Yugoslavia, were under Soviet and their supporters control. Poland, Hungary and Czechoslovakia were not so keen to work together with the SU and GDR, but it would have been exceedingly dangerous to travel into the Eastern Bloc. Chances were good that you have been caught immediately and extradited to the GDR.
Did the DDR formally revoke your citizenship and, if so, how quickly?
Lose one of the principal opportunities for punishing you by revoking your citizenship? Fat chance.
"Cold War" is not only a word, it was a reality. The leaders of both countries hated each other's guts. 17th July was a public holiday in the FRG and even in the 80s the teachers should mark the abbreviation "BRD" for BundesRepublik Deutschland (FRG in German) as error. Why? Because the "DDR" (GDR) uses it in print.