1) I have listened from a number of Shia people that burning door incident is weak, is that so?

2) if it’s weak, than why our scholars constantly repeat it from the mimbar?

3) Was Bibi Fatema martyred and it would be very much appreciated to provide me sahih hadees reference of her martyrdom

4) Was Hazrat Mohsin martyred proved from sahih hadith and it would be very
much appreciated if you provide me sahih Hadith reference

5) Do major Ayatollah believe in burning door and martyrdom?

She was martyred and the tradition is not weak. Of course all our Major Ayatullah believe in this. Check the link below for detailed answers:

http://www.askthesheikh.com/can-you-provide-reliable-shiasunni-sources-on-martyrdom-of-lady-fatima-al-zahra-a-s/