What is the meaning of "infallibility is obligatory for prophets and possible for others"? And is the muhaddath one whose heart is inspired with truth or one spoken to by angels? And what is the relation of this to accepting Abu Bakr's opinion regarding the ransom of prisoners over Umar's opinion, even though the Quran was later revealed in agreement with Umar's opinion?

The meaning of "infallibility is obligatory for prophets and possible for others" is that prophets are obligatorily infallible from major sins and are not affirmed in minor ones. As for others, it is permissible for Allah to grant them infallibility from sins and disobedience, but it is not obligatory.

And the Muhaddath is one who is inspired with truth and righteousness without revelation or infallibility. This can happen to some people, as it did to Umar ibn al-Khattab, may Allah be pleased with him. Ibn Hajar mentioned that the Muhaddath is the inspired one, into whose heart something is cast from the higher assembly, or one upon whose tongue righteousness flows, or one with whom angels speak without him being a prophet. What indicates this is the Prophet's saying, peace and blessings be upon him: "Indeed, Allah has placed truth upon Umar's tongue and heart."