4 is very natural.
1 would be natural if you left out the word "the" and wrote:
"I stay home barefoot all day long."
In 2 and 3, you need to add the word "at" and leave out "the:"
2: "I stay barefoot at home all day long."
3. "I am barefoot at home all day long."
Another possibility:
"I go barefoot at home all day long."
"Barefoot" means there is nothing at all on your feet. If you are not wearing socks, but you are wearing slippers or shoes on your bare feet, then you are not barefoot.
For some reason, the word is "barefoot", but it means that BOTH feet are bare.
I don't know any single word to describe having only one foot bare. If someone had one foot bare, I wouldn't say "she was barefoot," I would use a phrase or sentence to describe it--"she had a sandal on one foot, but her other was bare."
Completely tangentially, a nursery rhyme popped into my head. In this situation the child has stockings on, so he isn't barefoot:
"Diddle, diddle dumpling, my son John
Went to bed with his stockings on.
One shoe off and one shoe on,
Diddle, diddle dumpling, my son John."