I can answer the follow-up question: They were imprisoned according to paragraph 213 of the GDR criminal code.

First of all, I'm sorry if significant portions of this answer lack good, reliable English cites. I'll at least give English wikipedia links, for all the good they are, for some answers.

East Germans arriving in West Germany as refugees had one significant advantage over refugees from other nations. Since West Germany did not fully acknowledge East Germany, instead insisting on an exclusive mandate for Germany as a whole, the refugees were considered German citizens, with all rights. So, for example, they couldn't just be sent back because they had a right of residence in West Germany. So, in a way, the West German official view was that the arrivers did not get the West German citizenship, because they already had them. It was just officially recognized when they arrived.

However, here's where things get a bit more complicated and not as clear-cut. Especially in the early years before the border was closed off effectively, large numbers of East Germans relocated to West Germany. To channel those refugees, the West German legislature passed a law (the Bundesnotaufnahmegesetz, sorry German link only), which limited the right of abode for East Germans relocating to West Germany. In the following, several large central refugee camps were created. I would imagine, though I don't know for sure, that in those refugee camps, people were interviewed to (a) make sure that they actually were German, and not members of other nations, and to (b) to at least a rough prescreening for potential spies. This didn't always succeed. The maybe most famous example would be Günter Guillaume, who relocated to West Germany as a Stasi (the East German secret service) spy, and eventually managed to install himself as personal assistant to West German Chancellor Willy Brandt, leading to the Guillaume Affair and Brandt's eventual resignation. In any case, from those refugee camps, East German arrivers were eventually relocated to different areas of Germany. I assume that if they had family, they would be allowed to relocate there; otherwise, they would be distributed over the area of West Germany, so that no area had too many refugees to support. That support, at a bare minimum, would be the standard social security services available to every German in need. In addition, there seems to have been support from the Bundesausgleichsamt, which was created in 1952 especially with the goal of supporting the integration of German refugees arriving in West Germany. However, I cannot really tell you what amount of help they would receive from there; we're getting into a very specialized and obscure area here, and for a good answer would need an expert on that specific field of research, which I am not.

As to your final question, I can't say for sure, but I cannot imagine anybody being deported back to East Germany against their will. There were people who relocated from West to East Germany, and there were spy exchanges, but both of these were not deportations to East Germany against the will of those relocated.