I capped this and identified the movies I could. Thanks to foxxcub for the link so I could finally see this, since I missed it on Oscar night.
"Mind if I look at your Winchester?" Winchester 73. Awesome western B-movie.
I'm guessing this is Gunfight at the OK Corral, since that's Burt Lancaster.
Winchester '73 again, with Jimmy Stewart. It's a tale of brother vs. brother. Not slashy! But there is a lot of gun worship. ;-D
Red River with John Ireland (left) and Montgomery Clift. Now this IS slashy. They try shooting each other's guns. ;-D Very flirty.
"I'll have you spread-eagled over a wagon wheel." I'm guessing this is Fort Apache with John Wayne since his other cavalry film with John Ford, She Wore a Yellow Ribbon, is in color.
Paul Newman, probably in The Left Handed Gun as Billy the Kid
"Whatever happens, this is strictly between you and me. And the horse." The Big Country with Gregory Peck. He's a city slicker who needs to learn how to ride.
Gotta be James Dean in Giant.
Van Heflin (the farmer) and Alan Ladd (the gunslinger) work together to clear a tree stump in Shane
"I don't know why you thought you had to come say goodbye." Charlton Heston in The Big Country
"The goodbye that I had in mind would take a little more room than we have in here." Gregory Peck in The Big Country. The two then go outdoors and have an extremly lengthy fistfight, which is famously shot in long distance.