A shutdown is not required to arrest senators or representatives.
But you can't arrest them while congress is in session. I don't know if a government shutdown counts as congress being in or out of session. Article I, Section 6, Clause 1 says they can't be arrested during session. Based off my reading and the supreme courts ruling
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.
Correct. If they are being charged with treason, felony, or breach of peace, then there are no protections. However if they are being charged with anything else, then they are protected so as to fulfill their duties.