Using the US post office, is it cheaper to ship in a USPS flat rate box or use your own standard brown box?

Weight is the important factor. For a medium size Flat rate box, it might be cheaper to use another container if it weighs less than about three pounds. For the large Flat Rate box, about four pounds. Light stuff is always cheaper using some other container. Heavy stuff is cheaper in a Flat Rate box, if it fits. Books are always cheaper using Media Mail in your own container.

It depends. Why not try it both ways: get your own box from someplace or other, pack it, take it to the post office and ask them to tell you the postage. Then select the flat rate box that will work for your item and compare the prices.