Jamaat Salaah and Imaamat

Q: It is becoming quite common nowadays in many Musjids that late comers make their own jamaat in the sahen area of the Musjid. Kindly explain in detail the shar'ee ruling regarding this?

Q: While praying Salaah in Jamat (congregation) a person was standing far from the Imam and also there were no speakers. At one point the person thought that Imam Saheb said Allahu Akbar, and so he was going to ruku. As he was doing so he found that the rest of the muqtatis were still standing (the person in concern heard it wrong) so he again stood up. Now my question is that is the Salaah of the person, in concern, valid or should he repeat his Salaah?

Q: If for any fardh salaah, I performed it at home but made jamaat with my wife. Will my waajib obligation of jamaat namaaz be fulfilled even though the musjid was nearby?

Q: Please help answer this based on the Shafi'ee mazhab. If a musalli joins the congregrational prayer and he joins when the Imam is in ruku, will he still get one rakaat, how about al fatihah that he did not read for this rakaat?

Q:

1. If when following the Imaam, the Imaam says allahu akbar to go to sajdah and you say it only when he says it but finished before he did. Is the salaah accepted? An is this for all takbeer and salaam? 
2. If you proceed for sajdah only after the Imaam or same time but you are there before he is. Is the salaah valid?

Q: When I came late for salaah and had to complete missed rakaats I read durood-e-ibrahim and dua. Does this make sajdah sawhu waajib? If yes and previously I should do this out of ignorance and cannot not remember how many times are done it what should I do?

Q: The mosque I am going the imaam makes the witr as 2 rakaat followed by 1 rakaat in which he makes qunut after the ruku whereby everbody raise hands as in duaa and says everytime amin to the duaa.

If a Hanafi follows this imaam is his witr also valid? Secondly they start the tarawih rightaway after the fard of isha thus they dont perform first 2 rakaat sunnah. When should I pray this?

Q: I need to know if Hanafi's can read witr salaah in Ramadaan in Jamaat behind a shafi imaam? I need answers based on proof from the fiqh books.

Q: At home if my mum, wife and daughters make Jamaat, is there anything wrong? 

Q: Can a person making ruku and sajda follow an Imam making salah with signs in a nafl salah?