President-elect Donald J. Trump’s rapid move to stock his administration with loyalists, some viewed with skepticism even by members of his own party, is testing both his grip on Senate Republicans and the boundaries of executive power. Mr. Trump has made it clear that he wants the option of going around the Senate to install cabinet and other appointees without the chamber’s approval. He could do so with what is known as a “recess appointment,” which allows a president to act on his own when the Senate is not in session. But it is not clear whether Republican senators would...