I assume Obama wouldn't be responsible for anything that happened on the 20th January but neither Trump could be as he wouldn't have the chance to practically manage to take over the important contacts. Is there some kind of temporary representative both the former and the new president agree upon to run the country while they're switching their seats?
There is a website for presidential transitions, which says:
Inauguration Day marks the official end of the current Administration. In past transitions, the President has requested all non-termed political appointees to submit in advance a letter of resignation effective upon the swearing-in of the new President. On this day, agencies will begin on-boarding new appointees and will continue to do so over the course of 2017. On-boarding includes appointee orientation activities, such as briefings and workshops, which are funded through the GSA. On January 20, 2009, the Obama Administration had seven Senate-confirmed appointees ready to begin work.
In the past, some appointees have been asked to stay. For example, Barack Obama kept George W. Bush's Defense Secretary for several years and asked Bush's Secretary of Homeland Security to stay until the new one was confirmed. But that was by request, not something that happens automatically. Hillary Clinton took office the day after Condoleezza Rice left. Timothy Geithner took six days. Source.
The presidential portion of the transition is explicitly defined in the twentieth amendment:
The terms of the President and Vice President shall end at noon on the 20th day of January, and the terms of Senators and Representatives at noon on the 3d day of January, of the years in which such terms would have ended if this article had not been ratified; and the terms of their successors shall then begin.
There is no actual moment where neither person is president. The actual transition is seamless.
There is no lapse in coverage for executive power. President Obama will still be POTUS right up until the moment Donald Trump is sworn in on Jan. 20.
During a presidential election, the outgoing President and the new President-Elect form a joint transition team (as Obama and Trump have done) to handle the graceful transition of power from one administration to the next.
As for the executive officers appointed by Obama, they all remain in power until the new nominees are confirmed by the Senate.
However, I will add one important caveat. Technically, the official transfer of power happens on January 20. But as a practical matter people tend to begin cooperating with the incoming administration as soon as the election results become final.
You can see this in practice as the new cabinet members are appointed. And even the recent announcements of some notable manufacturers who have announced investment decisions to open plants or keep plants in the United States reflect this trend. In fact, on Obama's inauguration day, many of his appointments were approved by the Senate.
This trend is no accident. It is purely practical. Everyone realizes the outgoing administration will have no power after the inauguration and, therefore, those with an interest in maintaining a positive relationship with the White House tend to cede to the wishes of the incoming administration as soon as practicable.