Like Britain, we never have an interrex. Someone in the U.S. always has the power of the Presidency, right until the last Cabinet official is exhausted. And the Senate confirms candidates.
Not quite. The VP only becomes President upon the removal from office or death of the President. (25th amendment, section 1). According to the 20th amendment, section 3:
if the President elect shall have failed to qualify, then the Vice President elect shall act as President until a President shall have qualified; and the Congress may by law provide for the case wherein neither a President elect nor a Vice President elect shall have qualified, declaring who shall then act as President, or the manner in which one who is to act shall be selected, and such person shall act accordingly until a President or Vice President shall have qualified.
Clearly in the case of an ineligible, or unqualified, President elect the VP elect then acts as a President, but only temporarily.