The short answer: no and no, respectively.

Please let us know what source said that Chester Arthur left the Presidency early, in 1885. He didn't. The only presidential resignation was Nixon in 1974.

The question of presidental vacancy has had a few debatable moments over the years.

Though there was a United States before the Constitution, I would say that there was no President of the United States before the Constitution's George Washington. (Some suggest that the 14 men who had the title "President of the United States in Congress Assembled" counted, but that was the presiding officer of the continental Congress, he wasn't an executive, and he had little power even in the legislature.)

There has been a president continuously ever since.

There was a bit of debate at the time of the first presidental death, William Henry Harrison in 1841. John Tyler stated that he had become president. The Constitution, article I, section 3 (at http://constitution.findlaw.com/article1.html#1.3 ) is a bit ambiguous in the wording, with "in the Absence of the Vice President, or when he shall exercise the Office of President of the United States". Does "exercise" mean he was president, or just acting as president? The Senate eventually voted to accept just "President", and that precedent held in later cases. The question was put to rest in the 25th Amendment ( http://constitution.findlaw.com/amendment25.html ) with "In case of the removal of the President from office or of his death or resignation, the Vice President shall become President."

Sometimes the presidential term starts on a Sunday. Usually, the president takes the oath privately on the day, and then repeats it the next day at the public inauguration cerement. One time, the president refused to even take the oath of office on the "Sabbath". But the US Constitution ( http://constitution.findlaw.com/article2.html#2.1 ) just says "Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation ..." I don't know whether any legal scholars suggest that there was or is a vacancy in the office before taking the oath, but at least the vast majority would say that he was president on that day; he simply couldn't exercise the duties. (March 4, 1849; President Zachary Taylor. A common tale is that David Atchison was acting president on the day, but there are further objections to that notion.)

Mind you, there can be a vacancy in the office of President, as you note, if neither the residential and vice presidental offices are qualified and not vacant. It simply hasn't happened yet. Congress may by law provide for the case of there being no qualified President, "declaring what Officer shall then act as President" (meaning an acting president). The first law was the Presidential Succession Act of 1792, and it has been modified from time to time since. That act controlled in 1885, so if Arthur had resigned, the president pro tempore of the United States Senate would have been acting president.