The Spirit is what makes a person a child of God (Ro 8:14-16) which marks the beginning of Christian life. Baptism here does not mean water baptism but Spirit baptism at conversion. "We all were baptized in the one Spirit" and "We all were caused to drink one Spirit" are a piece of semitic parallelism (like in the Proverbs) where both clauses make essentially the same point.
In Christ, common distinctions, such as race and social status, have become insignificant.
All members of the body are needed; otherwise some function of the body would be missing.
The diversity in the body is by God's own design, repeating 12:11 that the Holy Spirit gives these various manifestations to "each person, just as he pleases". This theme is represented by the words: "determines"(v.11), "arranged" (v.18), "combined" (v.24), "appointed"(v.28). That explains why different gifts are present to form an optimal combination.
The eye and the head imply a view "from above", where those who consider themselves at the top of the "hierarchy" in the community suggest that they can get along without some others, possibly referring to the rich (following the passage on abuse of the poor in 11:17-34).
Paul has in mind the internal organs, which are full of "weaknesses", but are indispensable. These "apparent weakness" has no relationship to their real value and necessity to the body.
Parts of the body are apparently "less honourable" and "shameful" (unpresentable)--probably referring to sexual organs; yet we bestow special honour and greater docorum (special modesty, we cover them), while the more presentable parts such as the face do not have such a need.
Members of the body mutually need each other in order to function as a body. In 11:17-34, the division is precisely in terms of some members abusing others by not caring for their needs.
When one has a toothache; the whole body suffers with the part that is aching.
The first three in the list (of 8) do not refer to specific persons or office; they are ministries or functions or gifts. The attempt of NIV to turn the entire list into persons is unnecessary.
Paul intends the first three to be ranked, possibly pointing to their precedence in the founding and building up of the local assembly, not about relative importance or order of authority.
The list includes apostles (see notes for ch.4), prophets (Eph 2:20, 4:11), teachers, miracles, gifts of healings, helpful deeds (minister to the physical and spiritual needs of others), gifts of administration (more accurately "acts of guidance", not necessarily possessing administrative skills; although it likely refers to giving wise counsel to the community as a whole), tongues.
Paul is about to launch on his next argument (in 14:1-25), namely the need for intelligibility in the community; and in the community all the intelligible gifts are "greater" than tongues because they can edify while tongues without interpretation cannot. But he interrupts himself (an interlude in ch.13) to give the proper framework in which spiritual gifts are to function--love.
Paul does not call love a gift; he calls it "a way that is beyond comparison"--seeking the good of others before oneself. Love is not the greatest spiritual gift. Thus it is not "love versus gifts" but "love as the only context for gifts"; for without love, spiritual gifts have no usefulness at all--but then neither does much of anything else in the Christian life.