In the first scene after the opening credits, it's an outside shot of the Stockton Sheriff's Office, which shows the office having a double door entrance, with both doors having glass panes in them. When it switches to the inside, the door is shown to be a single wooden door. (In fact, it's the same set used as Marshal Dillon's office in Gunsmoke.)
The proof of the actual murderer was through comparing the weight of the bullets from the murder gun and Liberty's gun. The premise was that bullets of the same caliber from one gun will always weigh the same and be a different weight than bullets from another gun. In fact, the weight of a bullet is not determined by the gun it is fired from, but is determined by the manufacturing process. Bullets of the same caliber and manufacture (especially in the 19th century) may or may not have the same exact weight.
A newspaper and a wanted poster are shown with photographs printed on them. Photographs were not able to be printed in the time period that the show takes place. It wasn't until the 1880s or 1890s that photos were able to be reproduced in print.