Short answers, pulled from the story.
The Brunswick-Balke-Collender Cup was a silver loving cup donated to the American Professional Football Association by the Brunswick-Balke-Collender Company's Tire Division. It served as the championship trophy for the newly formed professional football league and was awarded only once, to the Akron Pros for the 1920 season.
The Akron Pros won the Brunswick-Balke-Collender Cup on the 30th of April 1921 at a meeting held at the Portage Hotel. The trophy was presented to team owners Art Ranney and Frank Nied by former Penn State star Timmy Bryant.
The Buffalo All-Americans and the Decatur Staleys disputed the 1920 championship because they had tied but never been beaten by the Akron Pros. Critics also pointed to a conflict of interest: team co-owner Art Ranney presided over the championship vote while the league's president and vice president were absent, and his own team was awarded the title.
The Brunswick-Balke-Collender Cup disappeared after being awarded to the Akron Pros in 1921. The APFA and NFL meeting minutes never mentioned it again, and it was never recovered. The only known visual record is a photograph that appeared in two newspapers from the era.
The Ed Thorp Memorial Trophy replaced the Brunswick-Balke-Collender Cup starting with the 1934 Championship game, thirteen years after the original was awarded. It was named for Ed Thorp, a referee, rules expert, and sporting goods dealer who died in 1934.
The NFL began awarding the Vince Lombardi Trophy to its championship team starting in 1970. Unlike its two predecessors, a new Lombardi Trophy is produced each year rather than passed from champion to champion. The trophy itself dates to 1967 and the first Super Bowl.