Matt is called a US Marshal. Kansas became a state in 1861, more than a decade before the series is set. There was only one US district court in Kansas and one marshal assigned to it, plus a number of deputy marshal. All deputies would be based in Hays, the capital, not towns like Dodge. And deputies would enforce federal laws and court orders, and capture federal fugitives. They would not have state or local jurisdiction (like breaking up fights in the Long Branch). At the time of the series Dodge had a town marshal, and a county sheriff with jurisdiction outside the town limits.
In the Old West, woman were either rarely allowed into saloons or never allowed into them. Women such as Ms. Kitty would only be allowed into the saloons if they were the wife or daughter of the owner and then only if they dressed in a sober fashion, rather than in colorful clothing.