On Diceman’s question, under Article I, Section 5 of the US Constitution, “a majority of each [house of Congress] shall constitute a Quorum to do business”.
I have some knowledge and resources about the Senate - and I would presume that the Senate is where this matters most (I don’t have the numbers in front of me, but I would bet that the withdrawl of Southern members had a far bigger percentage impact on the Senate than on the House.)
On a minor initial point, I know that under the rules of the Senate (and I suspect this is true in the House as well) a quorum is presumed to exist unless proven otherwise. So, any action taken by voice vote (or, according to Senate precedents, even by division vote) would stand regardless. However, a roll call vote can provide proof of the absence of a quorum. These votes could have been questioned.
Robert Byrd’s history, “The Senate 1789-1989”, does indicate that the seats of most Southern senators were declared vacant when terms expired, and those who supported the Confederacy were expelled as members - so the rules of the Senate would require a majority of all the seats. (A few members from non-seceding states were also expelled). I am pretty sure that the quorum would have been 35-37 members (West Viginia and Nevada were admitted during the war, so the Senate grew during this period.)
It is worth noting that some seats from the South continued to be filled. Andrew Johnson of Tennessee served until 1862. The rump legislature from Virginia appointed two senators from that state before they set up West Virginia. (I don’t know if the westerners continued to choose 4 senators, or just chose 2 after that)
In a quick scan through the Civil War chapters, I cannot see Byrd make any reference to quorum difficulties. My suspicion is that although this was a potential problem (the Republicans held 31 seats at the beginning of the war - the 10 Democrats in the Senate could have denied a quorum by refusing to show up), the political realities of the time made this tactic extremely hazardous for the remaining senators, and thus it was not used. Byrd was (and I presume still is) a master parliamentarian, and I would be stunned if his history would ignore this issue if it came up much. On the other hand, it could well be the case that part of the real reason for admitting West Virginia was to try to push the numbers of the Senate in a direction that made quorae (quorums?) a bit easier to acheive, and Byrd as a very patriotic West Virginian might have pushed that part of the story to the background.
I am guessing that any Constitutional amendments proposed during the war would have been treated as the immediate post-war amendments were - they would still have required a full 3/4 of all the states to ratify them, but newly created governments in the Southern states would not be recognized until they ratified the amendment(s).