Why indeed. What things can not happen if Congress is out of session or not even sworn in?

The are only two "official" branches of government right now.

Incoming House members who have not taken the oath cannot vote, engage in floor proceedings, introduce new legislation or conduct other official House business until a speaker is chosen.

What opportunities does this present? Arrests?

Article I, Section 6, Clause 1:

The Senators and Representatives shall receive a Compensation for their Services, to be ascertained by Law, and paid out of the Treasury of the United States. They shall in all Cases, except Treason, Felony and Breach of the Peace, be privileged from Arrest during their Attendance at the Session of their respective Houses, and in going to and returning from the same; and for any Speech or Debate in either House, they shall not be questioned in any other Place.

When is the House in session?

A new Congress begins at noon January 3 of each odd-numbered year following a general election, unless it designates a different day by law. A Congress lasts for two years, with each year constituting a separate session. The Legislative Reorganization Act of 1970 requires Congress to adjourn sine die no later than July 31 of each year unless there is a declared war, or unless Congress otherwise provides.

Sine die --> Latin for "indefinite"

What actions might the Executive or Judicial branches take at this time that would be prevented if the House were in session?