Those are good questions, and I really don't have the answers. From TSA's website (http://www.tsa.gov/who_we_are/what_is_tsa.shtm):
"We are the Transportation Security Administration, formed immediately following the tragedies of Sept. 11. Our agency is a component of the Department of Homeland Security and is responsible for security of the nation's transportation systems.
"With our state, local and regional partners, we oversee security for the highways, railroads, buses, mass transit systems, ports and the 450 U.S. airports. We employ approximately 50,000 people from Alaska to Puerto Rico to ensure your travels - by plane, train, automobile or ferry â" are safe and secure."
They are definitely acting on behalf of the federal government (as part of the DHS). I am not sure what recourse you'd have if they'd denied you access to air travel.
Whether they are trained or not in constitutional issues, that is irrelevant. It's up to you to be, as a free citizen, as you put it.
Walking out of any security related situation without permission is generally a bad idea (IMHO). Always ask permission ("Am I free to go?"). You can always sue later to attempt to redress an illegal detention.
Finally, you may want to look at some of these videos, and start your search from there:
http://video.google.com/videoplay?docid=8167533318153586646&hl=en
http://www.youtube.com/user/CheckpointUSA