Who presides over federal courts?

Judges preside over individual courts; the Chief Justice presides over the US Supreme Court, and a Chief Justice or Chief Judge (or similar title) presides over the state supreme court (or its equivalent).

All state judges are responsible for upholding their state constitutions, as well as the parts of the US Constitution that apply to them. Judges and justices in the federal court system take an oath to uphold the US Constitution. When conflicts arise between the two, the US Constitution is Supreme per the Article VI Supremacy Clause.

There is no single entity (including the President) who presides over the state and federal court systems as a whole.