How sound are the hadiths that indicate the revelation of Qur'anic verses in agreement with the opinion of Umar ibn Al-Khattab, and does their soundness imply the possibility for humans to produce something similar to the Qur'an, thereby invalidating the Qur'an's challenge to produce a surah like it?

The Quran is a miracle. Allah Almighty challenged the eloquent Arabs to produce a surah (definition) like it, but they failed. This challenge continues until the Day of Judgment, and no one can produce a surah, not even a short one.

As for what was reported about Umar ibn al-Khattab, may Allah be pleased with him, regarding his agreement with the Quran in some verses, it was an inspiration from Allah Almighty, not that he produced something similar to the Quran. It has been confirmed that Allah Almighty revealed a number of verses that concurred with Umar's opinion, may Allah be pleased with him, and this is from Allah’s grace upon him. He is among the "Muhaddathun" (those who are inspired with truth) into whose hearts righteousness has been cast.