Jamaat Salaah and Imaamat

Q: Some people were reading salah in the main hall and others outside the hall. The microphone got cut half way through, such that the people outside the hall couldn't hear the imaam or see him. There was also no one to call out the takbeer. What is the ruling in this case? 

1. Is the salaah valid for those outside despite the mic being off throughout the rest of the salaah? 

2. Some people broke their salaah and some carried on. Is their salaah valid? 

3. If the mic was to start working again, then can the people outside carry on and catch up to the imaam?

Q: If a lady went to the Haramain and she performed some salaah behind the Imaam, will she have to repeat those salaah?

Q: I want to stay home to focus on my studies and quit going outside because whenever I go outside the home, I get distracted. Can I perform my salaah at home rather than in the musjid?

Q: Do I have to makeup for salaah if it was led by a child who is not a mukallaf?

Q: Namaz behind such imam: 

If there's a Barelvi masjid in my area and they say the following before or after azan: "assalatu wassalamu alaika ya rasulullah wa ala alika wa ashabika ya noor Allah" 

Can I pray behind them and if not, should I repeat every prayer behind them?

Q: 

1. If the latecomer joins the Imam in the first or final tashahhud for even a moment, then does the latecomer finish his tashahhud before rising up (for either the third unit, or to complete his missed units)? If the Imam rises quite soon after the latecomer joins him in the first tashahhud, then is it a factor that the latecomer may be behind the Imam? 

2. A latecomer makes the opening takbeer while the Imam is in the first tashahhud. If the Imam begins to rise for the third unit while the latecomer is going down, what does the latecomer do? 

3. A latecomer makes the opening takbeer while the Imam is in his final tashahhud. If the Imam says the first Salam while the latecomer is going down to join him, what does the latecomer do?