No, you are not required to answer their questions regarding your travels or your citizenship away from the borde checkpoints. No, they may not search your car without a warrant or probable cause (being a pain in the ass..? Not probable cause.) Unless they are detaining or arresting you, you are not required to pull the vehicle over to secondary or exit the craft. No, you are not required to aid a law enforcement officer in any way that might remotely implicate you in a crime (particularly one they invent.)
A big problem is that these checkpoints are 20-40 miles inside the border. Citizens, obviously, live inside that border and might have to commute through the checkpoint. They have no right to impede your free travel.
This applies to more than the immigration stops: