Most of them. Nothing prevents the President from firing someone like Comey because the FBI reports through the Department of Justice. With few very exceptions anyone the President appoints, he can fire.
The Fed Chair is one of those exceptions. The Fed is supposed to be able to operate outside the political pressures of Congress and the President. Because of that the Fed Chair cannot be fired except for cause; illegal acts or misbehavior or something like that. Disagreeing with what the President wants does not constitute cause.
If Congress creates the position (like with all executive branch departments), it has the funding discretion as well as the authority to abolish the position within its Article I powers.
However, Congress cannot pass a law that curtails the President's Article II power to execute the law, right?
I think that Congress cannot create a position that reports under the Executive, and then declare that the Executive cannot exercise plenary Executive control over that position because that encroaches on Article II power, no matter how much they want to claim that it's a special exception.
Article II is supreme law of the land, and no bill that Congress passes to create a position is superior to Article II.
-PJ