How did presidential policy and supreme court decisions in the mid twentieth century United States contribute to a more involved national government?

I agree with the above answer, and I would emphasize that you need to define "mid-century."

To me, the rise in the power of the national government can't be discussed without linking it back to the Depression and WWII.  Both of those, more than anything after, led to an increase in the power of the central government.  As far as the Supreme Court goes, before Darby, there was the "switch in time that saved nine" where the Court started allowing New Deal programs to stand after it had been striking them down (this is West Coast Hotel v. Parrish).

After WWII, the federal government continued to do more or less whatever it wanted up until the Reagan years.  Even Republicans like Eisenhower and Nixon did a lot of things that increased and/or used the power of the federal government (building the interstates, aid to education after Sputnik, wage and price controls under Nixon).

So I would say that the Depression and WWII set a precedent and the need for the federal government to fight the Cold War added to that precedent.  People got so used to the federal government acting that they went along with it until the '80s.