I am not so sure what button-faced means in the first examples but it is clearly being used as an adjective. It might mean someone whose face tends to be helded scrunched up like a button. Perhaps it is a person who is normally quite stern -- or, as you say, plain.
The last example is the interesting one. Here button is the noun and stands by itself. Can you see that? Then it is modified by the phrase "faced with cork," that is, it has cork on its outside "face."
It can sometimes be very hard to tell just how two words go together -- as part of a two-word phrase, a noun modified by the second word, a noun and a verb (yes, "faced" could be a verb,) etc. I know that I have difficulty with that when I read in other languages; it is certainly not unique to English.
Thank you for the examples!