The show evolved over the years and was set in Nevada which wasn't touched much by the late unpleasantness so it's easy to disregard the timeline
Nevada did not become a state until 1864, so it is easy to see why it was largely untouched by the unrest. And it was admitted because of its large gold and silver deposits to help pay the bill for the party back East, even though it did not meet the population requirements.
Gun belts with belt loops existed before 1900, but they were uncommon because at the end of the Civil War there were very few cartridge guns, so there were no cartridges to put into the belt loops.
In spite of the numerous fans, Bonanza was a low-budget TV show about fictional characters living on a fictional ranch, it was never intended to be historically accurate. Guns used during the first few seasons were whatever the prop department had on hand, after that they could probably name their favorites and get them. But they still had to use the same blanks, same as John Wayne movies set in 1850s using 1892 Winchesters.