Article I section six of the Constitution protects member of Congress from arrest while they are in session except for treason or felony. The reason is so the executive can't just have them arrested because he doesn't like the way they are going to vote.

Yes, unless it is taken away by some rule, act or law. But, there is a caveat on this; that the immunity only exists when they are traveling to, or from, the capitol (generally speaking). It is interesting to note, that in many states this same immunity exists when a voter is traveling to, or from, a polling house (place to vote).

Yes, and they do not have immunity from arrest during Congressional sessions in this situation. William Blount was expelled from the Senate for treason in 1797.

exemption from taxation, searches, arrest, etc., enjoyed by diplomatic officials and their dependent families under international law, and usually on a reciprocal basis.

Yes for perjury or ig they are implicated with some new evidence. It's not double jeopardy if you were only a witness. Get immunity first if it's a concern.