I see I didn't answer at least one of your questions.

The flight crews would be civilian pilots simply because a) the military doesn't have all that many pilots qualified to fly the airliners and b) those they do have are already busy.

The airlines would certainly be compensated for the use of their planes. As the links stated, they already have financial incentives to contract their planes for CRAF services.

Quite a few airlines participate in the CRAF program. I think the second link provides a list of them.

The CRAF program has existed for decades but has only been used once for the Gulf War. During WWII, it was a different matter. Many civilian airliners were grabbed for military duty back then, especially the most capable planes. The airlines got them back at the end of the war.