Are the weaknesses in man intended by God? What are those weaknesses? Are they sinful?

None of those passages talk about weakness as sin, but it is about suffering and the general vanity of the mortal life. See Ecclesiastes 2 on vanity. Paul talks about the predestination language- the creation was subjected to vanity, not by itself but by God who suffered/allowed (not intended) it, in hope of redemption from the corruption and vanity to glory.

David Stern commentary:

19–25 Our inheritance involves an ecologically ruined world that will one day be restored (Ac 3:21, 1C 15:23–28, MJ 2:8–11, Rv 21:1). It was made subject to frustration because of human sin (Genesis 3:16–19)—the intractability of the physical world is not merely a natural law, and one day it will end (for an imaginative fictional portrayal of such a phenomenon see the final two chapters of C. S. Lewis’s The Last Battle, in his series, “The Chronicles of Narnia”). Of this we have a certain hope, for which we are waiting eagerly, but with patience. This is the larger context of whatever suffering, discouragement and doubt we experience—a context of hope.

In 2 Corin. 12:9, Paul talks about the suffering, and it can be applied to all general situations of frailty, anxiety, troubles, hardship we find ourselves. The power of God will be manifested the most when we are in the lowest. It is not about sin, which grieves God, from which he came to save us from by his sacrifice, to implore us to repent from sins and be reconciled to God. Sin grieves God, and repentance makes him rejoice. See Ezekiel 33, Luke 12.

[Isaiah 40:29-31 ESV] ²⁹He gives power to the faint, and to him who has no might he increases strength. ³⁰Even youths shall faint and be weary, and young men shall fall exhausted; ³¹but they who wait for the LORD shall renew their strength; they shall mount up with wings like eagles; they shall run and not be weary; they shall walk and not faint.

[2Cor 5:18-21 NHEB] ¹⁸But all things are of God, who reconciled us to himself through Jesus Christ, and gave to us the ministry of reconciliation; ¹⁹namely, that God was in Christ reconciling the world to himself, not counting their trespasses against them, and having committed to us the word of reconciliation. ²⁰We are therefore ambassadors on behalf of Christ, as though God were making his appeal through us. We implore you on behalf of Christ, be reconciled to God. ²¹For him who knew no sin he made to be sin on our behalf; so that in him we might become the righteousness of God.

[2Cor 12:20-21 NASB] ²⁰For I am afraid that perhaps when I come I may find you to be not what I wish and may be found by you to be not what you wish; that perhaps [there will be] strife, jealousy, angry tempers, disputes, slanders, gossip, arrogance, disturbances; ²¹I am afraid that when I come again my God may humiliate me before you, and I may mourn over many of those who have sinned in the past and not repented of the impurity, immorality and sensuality which they have practiced.

Man was made lower than the angels [Ps 8], and at least one specific flaw in man is that he is temptable, as Eve, and Jesus was. And that was present in the original sinless state of man before the Fall, i.e. to be temptable is not a sin, it is just a weakness, and you can even say man was designed as fallible. However Adam fell but Jesus didn't succumb to Satan's temptations.

Why was Man designed as fallible?

Perhaps so that Jesus can come as Son of Man, and one lower than the angels succeed where angels failed, and thus Satan judged.

Why do the Creation have to suffer for this heavenly drama and be pawns in this conflict?

We don't know exactly, but we know it is out of love.

For even any man would not sent anyone to represent him and to speak on his behalf but only his very own beloved is privileged to bear this burden.

And such is Man, the beloved of God.

And Man shall judge the angels, if that is any consolation for bearing all the pain in this life. We can meditate on Job again.

This answer is from the Reformed perspective (as requested in the tag) based on a 1991 sermon by John Piper, a Reformed pastor: Christ’s Power Is Made Perfect in Weakness. His sermon addresses 3 questions that match your questions:

1. What are the weaknesses that Paul has in mind here when he says, “The power of Christ is made perfect in weakness”?

2. What is the source of such weaknesses? Do they come from Satan or from God? Or both?

3. What is the purpose of such weaknesses? Is there a goal or an aim for why the weaknesses come?

Below are quotations from his sermon that answer your questions.

What are those weaknesses?

Answer: Insults, Hardships, Persecutions, Calamities. Quote from the sermon:

They are circumstances and situations and experiences and wounds that make us look weak; things we would probably get rid of if we had the human strength.

...

But in reality, we don’t usually have that kind of human strength, and even when we may have it, Christians don’t use it the way the world does. Jesus tells us not to return evil for evil (Matthew 5:38–42). Paul said in 1 Corinthians 4:12–13, “When reviled, we bless; when persecuted, we endure; when slandered, we try to conciliate.” And then he added, “We have become like the refuse of the world, the off-scouring of all things.” In other words, this kind of lifestyle, this kind of response to abuse, looks weak and beggarly and feeble and anemic and inept — at least it looks that way to those who thrive on pride and equate power with the best come back.

Are they sinful?

Answer: No. Quote from the sermon:

So you can see that what Paul has in mind here is not sin. He is not talking about a kind of behavior — like we might say he has a weakness for lust; or she has a weakness for overeating. Paul is not talking about bad choices that we make. He is not saying the power of Christ is perfected in my bad choices. Or, I will all the more gladly boast of my bad choices. Weaknesses here are not imperfect behaviors.

Are the weaknesses in man intended by God?

Answer: Yes. Just like how it was with Job, God permitted Satan (v.7) to harass Paul with a "thorn in the flesh" (v. 4).

... the source of our weaknesses may sometimes be Satan and his destructive designs for us; but always our weaknesses are designed by God for our good. This is why the truth of God’s sovereign grace is so precious in the midst of hardship and calamity. God is in control of Satan. Satan does nothing to God’s children that God does not design with infinite skill and love for our good. This brings us to the final question, which we have already answered.

Did Paul mean God's grace is made perfect in his weaknesses that are sins?

Answer: No, it's Christ's power that is made perfect. Also, as explained above, those weaknesses are not sins. In the sermon, John Piper taught that God's purposes are 3:

1. To pray to God for relief from Satan's purpose to harass us

2. To prevent us from pride and self exaltation. God thinks humility is more important than comfort or freedom from pain.

3. To glorify the grace and power of his Son Jesus (v. 9-10). God wants to make us a showcase for Jesus's power when we rely on Him not to escape from weakness but for us to trust God by faith while we are suffering, just like how Jesus trusted God while suffering on the way to the cross.