The mayor tells Walt to resign and that he is removed. Then the other sheriff comes in and claims that the mayor appointed him to investigate. The sheriffs are not appointed in WY so they have to be recalled and the mayor has no authority to remove the sheriff.
At the range the lawyer was shot the bullet would likely have passed through the back of the chair.
This episode, and most others, show the Absaroka County sheriffs wear six-pointed star badges...but, the patch on the sleeves of their uniforms show a 7-pointed star. In actuality, the star designs would match.