Wow! A lot of great choices here, but here are mine:
Fist Fight: Gonna go with “The Quiet Man.” The Duke was just toying with “that great oaf” until it was time to go home for dinner.
Gun Fight: Absolutely “Open Range.” Best gunfight ever put on film. It goes through phrases, like a sword fight. Each one is excellent, all together they are epic. “Heat” is almost as good, I will admit. But my second place choice would be going way back to Roger’s Rangers raiding the Abenaki village in “Northwest Passage.” Amazing camera work for the late 1930’s. And appears to have been shot in nearly real-time, starting from just before dawn to mid-morning. Truly deserves the term “classic.”
Car Chase: Think I have only ever watched part of one of the F&F movies, and was underwhelmed. Being old-school, I would go with “Bullit”, although “Ronin” is very good.
Don't you just love when Costner goes around the building and approaches the bad guy holding the damsel. He just walks up to him and shoves his gun towards him as he pulls. My now favorite Western.
Looking like I got my first choice of gunfight correct from all the replies. Heat is second or tied. Sure there are many others as we read on.