The constitutional language for appointments is:

He (the president) shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.

So the president can appoint various officials, but the Senate must, by majority vote, confirm the ones that Congress hasn't designated as not requiring confirmation.

On the removal side, there's this:

The President, Vice President and all civil Officers of the United States, shall be removed from Office on Impeachment for, and Conviction of, Treason, Bribery, or other high Crimes and Misdemeanors.

Note "all civil Officers of the United States". Any government employee can be impeached. A few judges have been impeached and convicted over the last 200 years.

That's all the Constitution says.

Cabinet members and some other high officials serve "at the pleasure of the President", and Congress has delegated authority for lower level civil servants to the executive branch and the Merit System Protection Board.

So the question for the various semi-independent boards and commissions is whether the president can remove them, or whether they need to be impeached to be removed. This is a real question where the members have a term of office set by law. Federal Trade Commission members have a 7-year term. Security and Exchange Commission members, 5 years. Federal Reserve commissioners, 14 years. Arguably, they should serve out their term unless impeached. The constitutional argument is that the executive branch has only enumerated powers, those listed in the Constitution. Since the constitution specifies both appointment and removal by impeachment, that covers the only ways such officers can enter office or be removed from it unless Congress provides otherwise.