In the Lord of the Rings (BOOKS)...?

yes they absolutely do - kind of like chicken wings though in the essence that they are there for functions other than flying.

The confusion is because on one line he states his enemy faced him and the shadow reached out like two vast wings.

then later: ... it drew itself up to a great height and "ITS WINGS" were spread from wall to wall. I don't know how he could make any clearer do you?

see the Chapter - The bridge of Khazad Dum.

Some people argue yes and others argue no.

Tolkien never actually states if they do or not. Wings are sort of described in the book, but some people think that the wings being described are more figurative than literal, while others think that they are more literal than figurative.

Yes