That’s incorrect. Article 1, section 4 states the first Monday of December as the date when a new congress would convene. This was normal for the colonies, as their legislative business was conducted during the winter, when there was little else to do.
It was the presidential inauguration date that was moved back to January 20th from March 4th.
No, you are wrong.
Until the 20th Amendment, new members of Congress were not sworn in until March 4 - Per U.S. Code Title 2 Chapter 1 Section 1 (http://www.law.cornell.edu/uscode/text/2/1). See the notes tab at that link for the change in that law required by the 20th Amendment.
The Constitution originally required Congress to meet at least once a year, in December. Since new members were not sworn in until three months later, that meant that the required December session of Congress often included lame duck members. The 20th Amendment changed the required Congressional session and the swearing in of new members to January so they were concordant instead of separated by three months.