The best way to find results is through the shows themselves. If it’s one of the bigger circuits, try mane-events or Mark Harrel horse shows. If it’s a smaller show, try the website of the state affiliate, for example, the Minnesota Quarter Horse Association has results from all their shows.
I don’t know of a list of horses that are qualified for the World shows. About all there willl be is a working order after the schedule is posted. I believe an individual horse’s record will show his world show qualifying history.
ETA: It looks like there will be a qualifying list under Competitor Resources when the Open and Amateur World Show page is added to the AQ website. Right now, the youth world and select world qualifiers are listed.