What James3.1 said, plus:
I like to link 1 Corinthians 12:22-25 with 1:26-31, because there are some commonalities in the two passages which go a long way in answering at least some of your questions.
Since the Corinthians had a problem with pride and were given to boasting, first about the leaders with whom they most closely identified (namely, Paul, Apollos, Cephas/Peter, and Christ), and second about their spiritual gifts (with tongues seemingly to many of them the "gift above all other gifts"), Paul needed to take them down a peg. He therefore addressed FIRST the partisanship based on their favored leaders.
The first three factions in the Corinthian church, namely, the Paulites, the Apollos-ites, and the Cephas-ites, did not realize, apparently, that as important as leaders may be in the smooth functioning of the local assembly, they comprise but one gifting of the Holy Spirit. For a group of followers to rally around one particular leader and begin to act as if their little clique is somehow superior to another clique simply because it comprises a different leader is unseemly. Why? Because Christ is not a divided body, but one body.
Since the ground is level at the foot of the cross, all such boasting about any given servant/leader contradicts the humble unity we are to exhibit in Christ. Moreover, the message of the cross puts to shame such factionalism and the supposed superiority which attaches to it.
In chapter one, then, Paul rebukes the Corinthians' boasting because it contradicts the essence of the message and preaching of the cross, which is all about humility, death to self, and the oneness of all those who freely admit their sins were what nailed Jesus to the cross. As for the fourth faction of believers in Corinth, the ones who claimed loyalty to Christ only, I suggest this group comprised the “super spiritual” believers who mistakenly thought their adherence to Christ alone meant they could ignore Paul, Apollos, and Cephas, not realizing the first spiritual gift God gave the church after Christ ascended to heaven was apostles!
When we go over to chapter 12, Paul addresses SECOND the pride and boasting of the Corinthians regarding their spiritual gifts. Paul has to remind them of the variety there is within the unity of the body of Christ. What unites all believers in giftedness is the bestower of all gifts; namely, the Holy Spirit. And while there is a variety of effects among the gifts, even within the same gift--since God makes only originals and not duplicates, God both bestows and empowers the gifts as He sovereignly sees fit. How, then, is boasting even possible? Clearly it is not.
Now here is where chapters 12 and 1 are connected. In chapter one, Paul asks the Corinthians to consider their calling:
". . . there were not many wise according to the flesh, not many mighty, not many noble, but God has chosen the foolish things of the world to shame the wise, and God has chosen the weak things of the world to shame the things which are strong, and the base things of the world and the despised God has chosen, the things that are not, so that He may nullify the things that are, so that no man may boast before God. But by His doing you are in Christ Jesus, who became to us wisdom from God, and righteousness and sanctification, and redemption, so that, just as it is written, LET HIM WHO BOASTS, BOAST IN THE LORD" (vv.26-31, emphases mine).
Notice the connections between the Corinthians' calling (viz., not many wise, mighty, or noble) and what God chooses to use (viz., the foolish, the weak, and base and despised). Paul is not trying to shame the Corinthians, but he seems to be pointing out diplomatically that according to worldly standards, the Corinthians, by and large, simply do not measure up!
Corinthians: not many wise; foolish
World: many wise
Corinthians: not many mighty; weak
World: many strong
Corinthians: not many noble; base and despised
World: many noble
Now look at chapter 12, where we read,
"On the contrary, it is much truer that the members of the body which seem to be weaker are necessary; and those members of the body which we deem less honorable, on these we bestow more abundant honor, and our less presentable members become much more presentable, whereas our more presentable members have no need of it. But God has so composed the body, giving more abundant honor to that member which lacked, so that there may be no divisions in the body, but that the members may have the same care for one another. And if one member suffers, all the members suffer with it; if one member is honored, all the members rejoice with it. Now you are Christ's body, and individually members of it" (vv.22-27; after which Paul lists spiritual gifts).
Do you think the contrasts in chapters 1 and 12 are merely coincidental? In chapter one, on the one hand, you have foolish-weak-base-despised-things that are not, and on the other hand you have wise-strong-nobles-things that are.
In chapter 12, on the one hand, you have members that are weaker, less honorable, less presentable, and that lack; on the other hand you have the more presentable members who, by implication judge themselves to be--because of their gifting--necessary (indispensable!), honorable, presentable, and lacking nothing.
In conclusion, the very people in the Corinthian church whom the world deemed foolish, weak, and base, and whom the fleshly believers in the church deemed weaker, less honorable, less presentable, and less necessary could very well have been the janitors, the garbage collectors, the shepherds, and the poop scoopers in the parade of the elephants(!).
Not only that, but perhaps a garbage collector was the best teacher in the church, or a janitor the best evangelist; perhaps a poop scooper was a top-notch elder (or under- shepherd), or a powerful prophet! In short, God's methods are so often countercultural, and yet, ironically, far superior to anything the world has to offer. (Compare Jesus' words about many of the first being last, and the last, first, in Mt 20:16,27; Mk 9:35; 10:31,44; and Lk 13:30.)