Good question!

The government resigned. The ruling party, the SED (German Socialist Party), was basically removed from power when their "right to rule" was removed from the Constitution, and all the party's leaders resigned. Then the CDU (Christian Democratic Union, a center left party which the current German Chancellor is part of) won the elections in 1990, and they oversaw the handover of power.

The industries (all classified as "VEB," which I don't remember the German for, but is basically "Publicly Owned. For example, if this was East Germany, we'd have "VEB Chrysler" and "VEB Pepsi") were handed over to a sort of trust organization by the government, which oversaw their privatization. Many were bought out by Western companies in the early 1990s, which is sort of why most German companies are in the West.

The Stasi (officially called the MfS (Ministerium für Staatsischereit, or Ministry for State Security), dissolved with everything else. Several members of both the Stasi and Grenztruppen (border guards) were persecuted, especially if they could be tied to any deaths. Erich Mielke, the head of the Stasi, was convicted of involvement in several different cases, and got six years. There were no mass persecutions of Stasi personnel as you might imagine; West Germany didn't have the resources to investigate and persecute everybody in the (very large MfS), so they mostly went after people suspected of serious crimes. There's still a lot of former Stasi people out there; many are still "loyal" to East Germany.

If you want to know more, I recommend a book called Stasiland by Anna Funder. It's a good read that will answer many of your questions more deeply.