Designated Hero: Young Indy is, let's be honest, a thief in the prologue, swiping the Cross of Coronado from the guys who found it because he thought he had better plans for it. Keep in mind Indy didn't know what Fedora and company (and their boss) were going to do with it (for all he knew, they were archaeologist-adventurers themselves!) — and even if he did, what made it his call to steal from them? We're presumably meant to be on his side, as when the film cuts to 1938...Indy's stealing the cross from Panama Hat and crew again.
And Panama Hat and crew end up dying as a result. Granted, it's implied that Panama Hat and his crew were criminals...
Fanon: For a while some fans thought (or hoped) that drinking from the Grail at the end gave Indy immortality, or at least an extended lifespan and slower aging. An older, slower Indy in Kingdom of the Crystal Skull ruined that idea. Then again, The Young Indiana Jones Chronicles shows that Indy lived to at least his early nineties in the early 1990s, so perhaps drinking from the Grail did give him an extended lifespan, if nothing else.
The scene where Indy flies and crashes the plane becomes a bit winceworthy when, in March 2015, Harrison Ford was injured when he crashed his plane, which was also a WWII-era plane. Not helping is the line "Fly, yes. Land, no..."
When Indy infiltrates the castle his dad was being held in, he tries to pretend to be a Scottish lord. When the butler sees through this he says, "But if you are a Scottish lord, then I AM MICKEY MOUSE!" Guess who acquired Lucasfilm in 2012?
Retroactive Recognition: Nick Gillard, who was the fight coordinator for the Star Wars prequel trilogy, is the Nazi who watches Indy fighting through the periscope, and is knocked out after he mocks him.
Win Back the Crowd: Temple Of Doom had a decidedly mixed reaction, with Steven Spielberg and George Lucas admitting they let their personal problems interfere with their work. They explicitly sought to go back to the roots of the first movie - return to a more light-hearted adventure, as well as bringing back such elements as Marcus, Sallah and the Nazis. By all accounts, the duo succeeded.