Article I. LEGISLATIVE DEPARTMENTThere are some interesting things here. There is an absolute use immunity for anything that a member of Congress may say during a debate. That is a legal immunity. The press is not restrained from asking questions. Only government compelled speech is restrained, as in court. A member of Congress could theoretically confess to murder in a debate and not be asked about that confession in court. The press could ask, though.
§ 6. Rights and Disabilities of Members
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.
No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been encreased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office.
Also expressly stated: no double dipping. Also, while in office a member of Congress may not be appointed to a civil post created or increased during that person's tenure. They could be appointed after the terms ends and the person is out of Congress, even for one day. I do not think a person could resign and be so appointed if the "time for which he was elected," (the term) has not ended, despite the early resignation.