What are three important events in Part 2 of Things Fall Apart?

Three important events that occur in part two of Things Fall Apart are Okonkwo's exile to Mbanta, the establishment and growth of the Christian community in Umuofia, and Okonkwo's suicide.

Part two begins with Okonkwo's arrival in Mbanta. His exile is, of course, a devastating blow to Okonkwo himself, the first serious setback in a life of brilliant success. However, it is also of vital importance for the narrative. Okonkwo is prevented from opposing the changes which take place in Umuofia, and the extent to which everything has changed in the community is emphasized by his returning as a virtual outsider after seven years.

The greatest single factor which undermines the structure of Umuofia's community is the establishment of Christianity, which challenges not only the clan's religious traditions but its social structure. By admitting outcasts, women, and men of low status, the church provides an alternative to the tribal hierarchy which at first seems insignificant, but becomes steadily more formidable as the church grows in size and power. The church's power increases as men of higher status begin to join, and education at the mission school is much in demand.

Okonkwo's suicide is the tragic climax of the story. It is not only a personal tragedy but a public event, symbolizing the collapse of a society. Obierika tells the District Commissioner what a terrible thing he is witnessing:

That man was one of the greatest men in Umuofia. You drove him to kill himself and now he will be buried like a dog…

The fact that the oblivious District Commissioner fails to see the significance of Okonkwo's death only emphasizes its importance in the eyes of the reader.

What are three important events in Part 2 of Things Fall Apart?

1. The exile: Okonkwo and his family move to his mother's village, Mbanta, to begin his seven-year exile for the accidental killing of Ezeudu's son. Okonkwo's family gives him a plot of land and the necessary seed-yams to start his own farm, but Okonkwo is deeply depressed and takes no pleasure in the work. He had been a leading man in Umuofia, with wealth and prestige. Circumstances have now forced him to start over, literally "from scratch," back with his mother's family as if he were still a child.
2. The white men in Abame: The first white man appears in the region, riding a bicycle into the village of Abame. The villagers are terrified; they consult their oracles, which warn that this man is a harbinger of destruction. The villagers kill the white man to ward off this evil, but more white men come, round up the villagers, and shoot them all—causing the exact destruction the villagers had sought to prevent. Okonkwo receives this news from his friend Obierika, who has come to Mbanta to visit with him. Neither of them knows what to make of the story, but both are sure that no such thing could ever happen in Umuofia.
3. Nwoye's conversion to Christianity: A few years after the events in Abame, while Okonkwo is still in exile, Obierika comes to tell him that white missionaries have arrived in Umuofia, and that people are beginning to convert to the new religion. While the consensus in Umuofia is that the converts were the "worthless" members of the community—low-status persons whose conversion is unimportant—Obierika was troubled to discover that Okonkwo's son, Nwoye, has also joined the missionaries. Given Okonkwo's previous status as a leading man of the village, Nwoye's conversion to Christianity is potentially much more disturbing to the community than the conversion of low-status individuals. On a more personal level, Obierika is dismayed on his friend Okonkwo's behalf. By choosing to align with the missionaries, Nwoye is rejecting the traditions of the Igbo and by extension, his father, who values those traditions. When Obierika confronts Nwoye about this and invokes Okonkwo in a sort of "what would your father think about this?", Nwoye explicitly rejects Okonkwo himself, saying "He is not my father." The unravelling of Igbo culture has begun, and is signified by this son rejecting his father.

What features of parts two and three contradict the first part of Things Fall Apart?

The universe punishes Okonkwo after the first part of the novel due to his hubris. He became so bent on appearing strong that he killed Ikemefuna, who was basically a son to him. Indeed, the participating event for the main character's banishment — one of unlucky chance, a rifle accidentally going off — can likely be read as a divine punishment evoked on Okonkwo.

So what lessons do the universe try to teach Okonkwo at the end of part one? That violence, power, and aggression are not the only forces out there — diplomacy, kindness, and calmness must also be exercised. So how does Okonkwo contradict these lessons he's taught?

The first way Okonkwo contradicts the lessons of part one is his rejection of his eldest son, Nwoye, after he joins the white missionaries. Nwoye acceptance of the new religion is a direct response to Okonkwo's killing of Ikemefuna, yet he refuses to bear any responsibility. Indeed, he resents the white missionaries so much it leads to his second contradiction: his extrajudicial killing of the messenger. Okonkwo once again believes his strength and power is enough to unite his people for war. But he is wrong, and unfortunately, the novel does not give him another chance. Okonkwo kills himself before he can be captured.

What features of parts two and three contradict the first part of Things Fall Apart?

I'm not quite sure what this question means, but many of the contradictions have to do with the character of Okonkwo and the arrival of the British. In these chapters, what were seen as "strengths" on Okonkwo's part (his masculinity, courage, prosperity, etc.) have now become weaknesses. He is broken by his years in exile, and the destructiveness of his anger is more evident than ever. His farm and crops are gone, and although he has 2 beautiful daughters ready to marry, he cannot initiate his sons into titled society immediately, as he intended. As he recognizes the threat the missionaries pose to the village, he speaks out, encouraging the men to go to war. Yet no one listens, and his frustration showcases how his bravery has become a liability.

In addition, solid foundations of the Ibo culture are cracking and splitting. Social structures are shattered by the acceptance of outcasts into the Christian church, and the converts grow bolder in their attacks on their native religion/society. This pits brother against brother, slowly eroding centuries of culture and civil structure.

Why are parts two and three shorter than the first in Things Fall Apart?

The length of the first portion of the book serves to establish the great power and tradition of the Igbo culture.  Over hundreds and hundreds of years they had built a narrative of the world and its creation and how they fit into the larger perspective of the world.  Achebe uses this longer portion to serve as the introduction to this powerful and established culture.

The shorter second and third parts are shorter in part to demonstrate the incredible speed with which things did in fact fall apart.  From the introduction of the Anglican missionaries to the massive conflicts between white colonists and black tribespeople just didn't take very long.  The precipitous decline of this powerful and celebrated culture happened so quickly, so too do parts two and three of the book.