When did Duryodhana know that Karna is actually one of Pandava's brothers and a son of Kunti (Kuntiputra) and how did he react?
In the copy of The Mahabharata full text I have, I do not see anywhere Duryodhana finding out that Karna was a Pandava (or Kunti's son).
That said, Smt. Kamala Subramaniam, in her translation of The Mahabharata (an abridged version, though substantially larger than other abridged & scaled down versions such as Rajaji's), states the following at the end of the Karna Parva.
Grieving Karna's death more than his own brother Duchasana's (which happened the same day shortly before the battle between Karna and Arjuna took place), Duryodhana goes to his grandfather Bhishma. He recalls how brilliant was the battle fought by Karna. He tells the grandsire anyone who witnessed Karna's battle would have thought he must have been a kshatriya. The Bhishma responds saying that was because he (Karna) indeed was a kshatriya. Taken aback by this statement Duryodhana asks Bhishma if the latter knew about Karna's birth and background. Bhishma says he knew long back, but refuses to tell Duryodhana, cautioning him that the truth could hurt him more. But Duryodhana persists, and then Bhishma tells him that Karna was Kaunteya. Shocked by that revelation Duryodhana asks Bhishma if Karna knew this, and Bhishma says yes. Duryodhana asks if Karna fought on his side despite knowing he was a Pandava, despite fully aware he was fighting his own siblings. Bhishma answers in affirmative.
Then Smt. Kamala Subramaniam recalls similar reference later when Duryodhana and Bhimasena prepare for duel.
After the 18th day afternoon, after the kaurava army was completely annihilated except for a few handful of folks (Duryodhana, Kripa, Aswattama, Kritavarma, the army of Yadhavas that came with Kritavarma, and Srinjayas), Duryodhana goes to a lake and takes rest. The Pandavas arrive shortly and challenge Duryodhana to come out and fight. During those exchanges, Yudhishtra advises Duryodhana to wear armour, and take his favourite weapon. Seeing this gesture of The King Yudishtra, it seems Duryodhana felt that righteousness was in Pandava's blood, and a common quality of Karna and his younger brother Yudishtra.
Now that said (summarized from my memory as I don't have the book right in front of me), Smt. Kamala Subramaniam did not mention the source. She did not mention which recension of The Mahabharata she used for her translation.
I would not be quick to assume she made up this whole thing in her mind, as that is not a respectful thing for an author who took upon the task of writing an abridged version of this great epic. So I will give a benefit of doubt assuming she must have found some recension that does mention these incidents. If it is so, then I don't know which recension it is, as I have not come across one in my search so far.
The issue is there are a few recensions and editions of The Mahabharata, and it is hard to find coherent descriptions of the events. Even the edition I have has repeating incidents across the chapters, same characters reappearing, not to mention print errors etc. So to sum it up -
1) None of the recensions that I consider to be early, or possible off springs of what might have been close to original, carry any reference to Duryodhana finding out about Karna's true identity.
2) Smt. Kamala Subramaniam mentions those incidents, which, if traced to some authentic version, could go to prove that Duryodhana found out about Karna after Karna's death.
Here is Smt. Kamala Subramaniam's book, published by Bharatiya Vidya Bhavan. Check the last chapter of Karna Parva, and also the chapter that narrates the battle between Bhimasena and Duryodhana.
Duryodhan never knew that Karna was the son of Kunti, because Kunti revelaed this truth after finishing the war means Duryodhan was already dead when kunti revealed the truth to pandava.Source [https://www.youtube.com/watch?v=Ll2MVIZONgg]