In what way (if any) was Adam's physical body and mental/spiritual state altered when God removed his rib to create Eve (Genesis 2:21–22)? Was there a specific reason the rib was removed?
In actuality, many meforshim read the pasuk to mean that Adam was created hermaphroditic, both "male" and "female". The separation is often translated as "side" (see Rav Hirsch or Art Scroll on 2:21) rather than rib and means that he was separated into two completely different beings "because it is not good for Man to be alone" (2:18). The purpose was to teach Adam that he did not have a companion and that he needed to have the "ezer knegdo" who would enable him to become "good". He needed to see and understand this as well as also seeing that both aspects were part of the original singular creation. Rav Hirsch explicitly states that this shows the complete equality of men and women.
21 And the Lord God caused a deep sleep to fall upon man, and he slept, and He took one of his sides, and He closed the flesh in its place.
You have formed me before and behind (Psalms 139:5)… R. Jeremiah b. Leazar said: When the Holy One, blessed be He, created the first adam, He created it with both male and female sexual organs, as it is written,
Male and female He created them, and He called their name adam, (Genesis 5:2). R. Samuel b. Nahmani said, “When the Holy One, blessed be He, created the first adam, He created him with two faces, then split him and made him two backs – a back for each side.” (Genesis Rabbah 8:1)
There is an opinion in the Talmud that states that God originally created Adam as a hermaphrodite and then split that one being into two separate bodies. Besides the verse that you cite, there is another a bit further on in Genesis that alludes to this concept: “Male and female He created them, and He blessed them and called their name Adam on the day they were being created.” (Genesis 5:2)
In what way (if any) was Adam's physical body and mental/spiritual state altered when God removed his rib to create Eve (Genesis 2:21–22)?
His physical body was altered that he now had flesh where a rib had been. (This is because bone doesn't grow back when removed. Haamek Davar.) I don't know about his mental or spiritual state; hopefully, someone else can answer that.
Was there a specific reason the rib was removed?
To create Eve from.