When Joe's plane clips the tree and loses the right wing, the plane rolls to the left. In fact, asymmetric lift would roll the aircraft to the right due to greater remaining airfoil surface on the left side.
When Carter is shot by Jean Arthur's character, the hammer on the the revolver had not been drawn back. A revolver will not discharge from just being dropped on table unless it had been cocked; and even then it's not a certainty.
When Cary Grant enters his room and unknown to him, Jean Arthur is taking a bath, there is a coffee pot brewing on the hot wood burning stove. Arthur places the pot on the table. The "steam" coming from the hot pot is not steam. Steam rises. To simulate hot steam, it appears to be dry ice in the pot as the vapor falls to the table and an over abundance of it.
Toward the beginning of the movie, when Tex the lookout radio guy says, "OK, it's open," you can see the whole mountain range in the background slightly shift to the right. (Apparently, somebody was moving the set backdrop or bumped into it while the scene was being filmed.)
When McPherson takes off with the patient, the chimney smoke shows he's taking off with the wind instead of against the wind. Taking off against the wind would increase his relative speed and help him take off.