If they DID send the guard to Chicago and a court limited them to what they had in LA (guarding federal buildings, protecting federal officers), there are like 5 big Federal buildings and no border to guard, so they'd REALLY be twiddling their thumbs here.
Sadly Trump can easily override those constraints thanks to the little known "I do what I want" loophole
It depends. They put the restrictions on them in California and they (mostly) obeyed them. It wouldn't surprise me if the same pattern happens elsewhere (district judge temporary restraining order on the whole activation without permission from the governor, stayed except for limiting their scope)
I can assure you that nothing is going to look like government waste more than two dozen bored guys standing guard at each of our 168 post offices.