I do not understand this question. It is a Mishna in Berachos 20 that clearly says they are chayav in mezuzah.
Correct.
This Halacha is also brought down clearly in Shulchan Aruch Yoreh Deah 291-3, as well in Rambam mezuzah 5 -10.
Also correct, but that doesn’t mean no rishon holds differently (which is highly unlikely anyhow, since he would have to find a diyuk in a different mishna, which argues on the one you mentioned from Berachos, and pasken like it).
What’s the mitzvah of mezuza?
To kiss it as you walk thru the door?
To attach it to the doorway.