No evidence of any such transfer exists. Only on a movie did this ever happen.
Not sure about the draft card; never heard that before.
As for stashing it...I'm guessing seeing someone walking around with a rifle after the President was shot with a rifle would lead some to believe that he shot the President with said rifle. But that is just a hunch on my part.
To my way of thinkign, having the cop see you on the 2nd floor (without a rifle) drinking a soda pop and then slipping out after the cop went upstairs (exactly what Oswald did) would seem to be exactly what you would do if you were expecting to get a ride to the airport or safehouse. And, to go further, taking various forms of transit to the Texas theater suggests that he was going to meet someone.
I don't know why you think that is an "AH_HA!" item.