The playing of a 1927 song in the 1940s ("Ain't She Sweet") is not an anachronism. Playing a 1940's song in 1927 is. "Ain't She Sweet" has been recorded dozens of times since it's introduction, up to 1990.
When Adam Trask is being shown how to operate the car he is going to buy, the mechanic takes Adam to the front of the car to show him how to use the manual hand crank. You can see the boom mic in the reflection of the cars front windshield.
Near the beginning of the movie after Cal climbs up on the train, the position of the sleeves around his neck changes between shots. Also the long shot shows him covering his arms with the jumper, but in the close shot his arms are still outside.