They did, and they carried out their normal duties. Many important events took place in the U.S. Congress during the course of the war. Nevertheless, not all Congressmen went back to Washington. The states who seceded were eventually ruled to still be states within the United States, but their senators were expelled. Until the states could be brought back into the Union their seats would remain empty. Some of the seceded states still operated a kind of government in exile, as in the case of the Wheeling legislature of Virginia (later to become West Virginia) or as in Union strongholds in Missouri or Kentucky that were split. These exiled governments were typically not allowed to seat Congressmen because they were elected by such a tiny minority of the voters, in some cases only a few dozen people casting votes. To allow them to fill the vacated seats of the seceded and expelled members would have made a mockery of representative democracy. Instead they were allowed to observe and comment in many cases, but not sit or cast votes. When West Virginia joined the Union in 1863, they were allowed to seat their Congressmen. As the former Confederate states rejoined during Reconstruction they were allowed to seat their Congressmen as well.

Yes, Congress did meet throughout the entire war. While throughout the war, Republicans controlled Congress, there was a large anti-war/anti-Lincoln portion that attempted to thwart Lincoln's war aims the entire war. They continued to pass laws, allocate money for the war, etc. They also set up the Joint Committee on the Conduct of the War to investigate early Union losses, mistreatment of soldiers, among other things. So in short, Congress functioned as usual, just with less people as the Southerners went home to form the Confederate Congress. I have done research on this due to papers I have written in a university civil war history class.