At the end of the show, Jake is in his barber shop and over his head you can see 3 rifles on the wall. Shouldn't those rifles be in the possession of Sheriff Matthew, as Colorado Springs has a gun law in this episode?
This is incorrect. The townspeople are allowed to OWN firearms and use them to protect their property. No one is allowed to CARRY firearms in town.
Horace complained that Myra stopped writing letters but previously she admitted to Hank that she could not read. Correction: Early in the series Myra could not read. Horace has since taught her and she even helped him with the telegraph.
The scene where Jake has locked himself in his shop and his Pa comes knocking on the door. You can see a light shade in the opposite window.
Dr. Quinn has a very sentimental photograph in one of the clinic's recovery rooms. It is unlikely that a doctor would have something so personal and special in a room that could be used by any patient. There is always the possibility of it getting contaminated and needing to be burned.