If no candidates get a majority in the U.S. electoral college, are the President and Vice President chosen by the incumbent Congress or the ones just elected?
The newly elected Congress does all of the work in electing a new President.
Under the 20th Amendment, the newly elected Congress takes office on January 3. Then three days later, on January 6, 3 USC § 15: Counting electoral votes in Congress, requires the new Congress to meet in Joint Session to count the electoral votes.
If this session does not produce a President or Vice President, there is what is called a contingent election. In a contingent election the House begins immediately to choose a President from among the top three electoral college vote getters, while the Senate chooses a Vice President from among the top two electoral college vote getters. Both Houses use majority rule. The House votes by state, so a majority is 26, while the Senators vote individually, so a majority is 51.
If the House does not pick a President by Inauguration Day, January 20th, the Vice President serves until a President is picked. If neither a President nor a VP has been picked by the 20th, the Presidential Succession Act applies, and the Speaker of the House, President pro tempore or a cabinet officer serves as Acting President.
It wasn't always done this way:
The 20th Amendment was passed in 1933 to take control over elections away from the lame duck Congress. Before the 20th A was adopted, the terms for P, VP and Congress all ended on Inauguration day, March 4. That meant the lame duck Congress had to deal with electoral matters. By giving Congress and the P/VP different expiration dates, the Amendment meant new Congress could deal with the election. Setting the election counting date after the new Congress was seated (on January 6), meant only the new Congress could.