Is it allowed in Islam to critically analyze the lives (events and happenings etc, discussed in the Quran, hadiths or accepted as historical facts) of Prophets, specially Muhammad (Peace be upon him)? If it is allowed are there any recommended approach, cares and precautions specified in Islam?
In a sense, your question doesn't make sense. Allah says:
Translation: Indeed in the Messenger of Allah (Muhammad SAW) you have a good example to follow [...] (Surah Al-Ahzab, verse 21)
And in Surah Yusuf:
Translation: There was certainly in their stories a lesson for those of understanding. [...] (Surah Yusuf, verse 111)
These verses make it pretty clear that the lives of the prophets (all of them) are here for us to learn and benefit from. And there is no way that we can accomplish that without critically analyzing their lives an extracting lessons from them.
This would include things in the Qur'an, as well as books of ahadith.
However, there is one caveat; rasulullah said:
[...] Do not prefer me over other prophets (like Moses) [...] (Saheeh Bukhari)
That is to say, we do not say that any prophet is better or worse than any other. This is despite rasulullah also saying:
[...] I will be the leader of the children of Adam on the day of resurrection [...] (Saheeh Bukhari)
Also, prophets need to be treated respectfully, because they are divinely sent by Allah -- even if we don't always understand everything about their lives.
According to Shia Islam yes. There is no limit about asking any question about God, any prophet, any wife of prophet, any Caliph, any war or any of 12 Imams of Shia.
Only if contains insult it has problem and it is recommended the asker first increase his knowledge about their life.
Asked from online answering service of Official website of Grand Ayatollah Makarem Shirazi