"To set foot" or "to set feet"?
North America.or the North America.

1. It is generally recognized that Leif Ericson was the first European explorer to set feet in the North America.

2. It is generally recognized that Leif Ericson was the first European explorer to set feet in North America.

3. It is generally recognized that Leif Ericson was the first European explorer to set foot in the North America.

2. It is generally recognized that Leif Ericson was the first European explorer to set foot in North America.

Question

"To set foot" or "to set feet"?
North America.or the North America.

1. It is generally recognized that Leif Ericson was the first European explorer to set feet in the North America.

2. It is generally recognized that Leif Ericson was the first European explorer to set feet in North America.

3. It is generally recognized that Leif Ericson was the first European explorer to set foot in the North America.

2. It is generally recognized that Leif Ericson was the first European explorer to set foot in North America.

Answer

It is generally recognized that Leif Ericsson was the first European explorer to set foot in North America.

'Set foot' is a fixed expression.
Names of continents do not take articles.

Answer

set foot, not set feet
North America, not "the" North America.
So your fourth sentence is correct.

Answer

It is always to set foot, unless you are something like ' I set both feet down so the doctor could examine my broken toes ;