First, I didn't realize that the US government's role was to act as an advocate for specific private plaintiffs in civil cases. Where in the Constitution do you find that?
Second, a top team volunteered to help the Schindlers. They turned them down flat.
Didn't know that. Who?
As for the other, it is certainly not uncommon for the federal government (state governments too) to enter briefs in cases to which they are not a party. In fact, I'd say they have a duty to if there are public interests at stake as in this case.