At a minimum, since this would likely necessitate some overlap where the person has these two jobs at the same time
This doesn't necessarily follow. Nomination and confirmation first; resignation followed by an oath of office afterward. No overlap. Note that a similar situation exists for members of Congress when they shift positions and they are in fact legally precluded from holding a second office.
is it permissible for any single person to receive multiple paychecks from the Federal government on a given payday?
Perhaps for any other employee of the executive branch, but not the President. From Article 2, Section 1.7 of the US Constitution, the domestic emoluments clause:
The President shall, at stated Times, receive for his Services, a Compensation, which shall neither be encreased nor diminished during the Period for which he shall have been elected, and he shall not receive within that Period any other Emolument from the United States, or any of them.
So no, the President can't take a paycheck as a Supreme Court Justice while serving as president. Whether that is sufficient to require the president's resignation in order to accept the Supreme Court Justice position is unclear. No one has ever tried it. The only president who became a Supreme Court justice was William Taft who was appointed three terms later by his successor's successor. Not even a hint of overlap.
Unlike with Congress and the President, there are no explicit constitutional rules limiting a Supreme Court justice from holding other jobs. Nor do their oaths have explicit references. It is possible that other legally applicable rules exist in their employee handbook or similar. Note that as a practical matter, all Justices have been required to give up other employment. For example, Elena Kagan had to resign as Solicitor General to become a Supreme Court Justice.
The domestic emoluments clause may be sufficient. It's certainly a peg on which the Supreme Court could hang a ban on holding both jobs. And there's certainly sufficient room to argue that it is not sufficient. It's a judgment call that could go either way.