Let me first try to answer your questions 1 and 2:
First, you have to separate between the different branches of the Stasi. There were the domestic spying departments and the foreign espionage depertments. (see Wikipedia for some introduction).
For the foreign espionage departments, the Hauptverwaltung Aufklärung (HVA), the recruitment was like with any other intelligence agency, including selection of promising candidates, background checks, and so on. The reasons for joining this part of the apparatus were also similar: patriotism and/or ideology, (some) adventure, and money.
However, I assume your question is more about the domestic spying agency, and especially its Informelle Mitarbeiter (IMs; basically informers). Here, the motivations varied wildly.
Some people did it for ideological reasons. Some of them were later used to inform on ever more harmless subjects (and close friends or even family), but couldn't/wouldn't stop or go public because they feared repercussions.
Others did it simply for money (as most informers do).
Often, young people were approached at a time when they
- were fresh out of school, and still very indoctrinated
- wanted to start studying, performing, or traveling and could be lured with support and approval
- were in the (mandatory) military service
- feared repercussions from something their friends or family members had done (like, setting up an antenna to receive western TV or similar "crimes")
Now to your third question:
Especially in the beginning, many people saw the GDR as a real chance at doing things differently. The Kaiser had been removed, the Nazis had plunged the world into war and murder and in the process devastated Germany, and the business leaders had always participated. The communists had been the only truly outspoken opposition for 50 years and had basically been proved right. Of the first Government members, some had fought Franco in Spain in the 30ies and spent the war in KZs or Gestapo jails, lending them credibility on the anti-fascist stance.
As Stalinism showed that there was murder and injustice in that system, too, many people became disillusioned.
But still, many (even most of the oppresed opposition) hoped to find a way between die-hard-communism and unrestrained capitalism. That's why many still balk at having the GDR described as "communist", when they would have liked to see it as "socialist" (There's a big difference between the two!).
Nobody wanted Soviet-style communism, and the leaders gave up on that in the sixties. But still they tried to control everything (including where people could tracel, what they watched, who could go to the university etc.), to stifle any dissent and meaningful discourse on what the stated goal of "socialism" should actually mean.
Many people thought it was a good idea gone wrong, and could and should be done right instead of abandoning it.
Stuff people generally remember positively: work for all, no homeless, no one hungered, medical treatment for everyone (although not up to western standards), good education system for everyone, solidarity, the (professed) love for peace, success at sports.
What many people don't remember (or try to forget): Stasi, pollution, a general sense of confinement, lack of opportunity beyond working a good job, lack of high-tech consumer goods (mostly because of the CoCom embargo), doping.