I read these stories maybe 30 years ago so I am a bit fuzzy. I am sure there were other stories in the book and I thought for a while they were in "The Illustrated Man" but I checked and couldn't find them there. In one story a mature driver is harrassed by a young lady in her new VW Beetle (maybe) and he lets her know to back off by pressing a button reveal missiles concealed behind his taillights. There was a line in that story that was something like "some cars were more equal than others", which I thought was cool and often remember. I don't remember if he offs here. In another of the stories, which I think was called "Friday Night Lights", a sport very much like American Football is played but the players are armed with real weapons like hand grenades etc. Any help would be much appreciated.
If Revenant is right that the one story is Alan Dean Foster's "Why Johnny Can't Speed", then there's a decent chance that you read them in The Science Fictional Olympics, edited by Isaac Asimov, which also has "National Pastime" by Norman Spinrad.
"Why Johnny Can't Speed"
Relevant quotes from story:
There were a few—un-American dirty commie pinko symps, no doubt—who decried the resultant proliferation of "argumentative" devices among high-powered autos. Some laughable folk even talked of an "arms" race among automakers. German cars made their biggest incursions into foreign markets in decades. Armor plating, bulletproof glassalloy, certain weaponry—how else did those nuts expect a decent man to Drive with Confidence?
He blasted the horn once and the frantic driver of the heavily neutral-marked vehicle made haste to get out of his way. Theoretically all cars on the surface streets were equal. But some were more equal than others.
"National Pastime" - Review
In the long-winded and somewhat renowned “The National Pastime,” Norman Spinrad goes into painstaking detail describing – from the point of view of the cynical and failed filmmaker who invents the game as a means of advancing his career with a television company – Combat Football.
Even less logically, fan casualties and fatalities are included in the statistics, with numbers soaring into the hundreds per game. They are played in stadiums without security. The plays are portrayed somewhat realistically as akin to American football, but the key play consistently seems to be a punch in the mouth.
The book is likely Alan Dean Foster's short story collection "With friends like these".
In the car story, there is excellent public transport in California which any sane person uses. Roads are effectively a war zone with heavily armed cars, were you go for adventure. A young man gets killed on the road, and his old man goes on the road for revenge. The young lady in the Beetle is chased off the road by him so she can't get into any really dangerous situation.
There is a possibility you may have read "Killerbowl", by Gary K. Wolfe. As I recall, the playing field was an abandoned city block, the game lasted 24 hours, the football players wore body armor and were equipped with melee weapons and the player who was the safety was armed with a rifle with one bullet...
Another option is the short story "The Survivor", by Walter F. Moudy. A battalion of United States Army soldiers up against a Soviet battalion at the 2044 Olympics.