Throughout the book Dr Manhattan is shown to be disinterested in human affairs. Veidt describes him as apolitical. So why was he seemingly a pawn doing the US governments bidding for so long before the main events of the novel? He obviously has the power not to do what he's told.
Immediately after his transformation to Dr. Manhattan, he continued to do things he would have done as a human. He continued his relationship with Janey, for example, even though he no longer seemed attached to humans, or at least, not in the way that humans are. Similarly, I'm sure he had the idea that when he was a human, if he had had the abilities he had as Dr. Manhattan, he would have used it to help his country. As he continued on, he became more and more detached from his former humanity. He came to terms with the fact that he was no longer human, and started to sever his ties to humanity. But at the same time, he was always apolitical, in that other than for the memory of his former attachments, he didn't care one way or another.
Because he was working for the US military studying nuclear physics; and after the accident, the military marketed him as a government superhero
To answer your second question: He slowly lost interest in human affairs; his relationship with the second Silk Spectre demonstrates that... And the Comedian also stated the Dr. was losing his humanity during the Vietnam War
He also apparently perceives the past, present and future simultaneously and that could also explain why he doesn't get involved, because he already knows how it will turn out.
and to answer your last question: Before the accident he always did what he was told: his father forced him to become a physicist; the military told him what to do and Slater was his handler after he became a superhero. Then he eventually went on his own...
Also too, remember that Dr. Manhattan has no free will. The past and the future are the same thing to him. All he does is step through time following a script, basically. In other words, he's fated to do whatever he is doing. There was an exchange in the book where they go into that.
Laurie Juspeczyk: Is that what you are? The most powerful thing in the universe and you're just a puppet following a script?
Doctor Manhattan: We're all puppets, Laurie. I'm just a puppet who can see the strings.
Dr. Manhattan is a creature of physics. His life is like a marble rolling down a hill. The marble can't suddenly decide to roll uphill - physics dictates that it has to roll downhill. Likewise, Dr. Manhattan can't simply "do something else", he has no choice. His future has already been written. He does what he does because he has no ability to do otherwise.