Does the Justice Dept. have to agree to Fast Track?
The Supremes can reach down and grab it if they want to.
I don't know.
That depends on whom you ask. According to Senator Schumer, the three branches of government are the executive, house, and senate. In actuality they are the legislative, executive, and judicial. As a separate branch of government, they get to make their own decisions about when they schedule cases and whether to "fast track".
No, although they will have opportunity to file briefs on the matter, and the Court will take those briefs under advisement, the State doesn't enjoy any kind of "veto" power.