Does a J Stevens 410 shotgun stamped with 'patented August 12 1913' mean it was manufactured before 1920?

You have the date of manufacture pretty close. No earlier than the 1913 patent date, and actually before 1916. New England Westinghouse purchased the Stevens plant in 1916 and used it to make Browning machine guns and Mosin-Nagant (Russian) military rifles. In 1920, the facility and Stevens name were purchased by Savage Arms Co. There is a lot of information on the Stevens company because it was one of the earliest US manufacturers of cartridge firearms (established in 1864 I believe). Stevens made several models of single shot shotguns, but there's not much collector interest in them, so there's not a lot of specific information available. Some models are probably scarcer than others, but most references just lump them all together and say they were produced from 1864 to 1948. Even in very good condition, most won't bring more than $75, but a 410 may bring twice that much.