1. "Cheeseburger" is a single word
The menu is your best guide: if it says on the menu: "1/2 lb cheeseburger" and "1/4 lb cheeseburger", then you would say,
"A half-pound cheeseburger, please" / "A quarter-pound cheeseburger, please."
or
"Cheeseburger please, a half-pound one." / "Cheeseburger please, a quarter-pound one."
Sometimes burgers are called half-pounder and quarter-pounder, in which case, "Cheeseburger please, a half-pounder." / "Cheeseburger please, a quarter-pounder."