Masbooq and Laahiq

Masbooq starting his salaah while the imaam is in qa'dah or making salaam

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?

Questions related to a masbooq

Q: Kindly answer below questions related to salaah: 

1. If a muqtadi misses some rakaat, at what point should he/she leave the jamaat and stand to continue the missed rakaat? Should he/she do it after the imam completes the 1st salaam or after the second salaam? 

2. For a muqtadi who has to missed a rakaat in jamaat salaah, should he remain quiet after the last tashahud behind the imaam and wait silently for the imaam's salaam to continue the missed rakaat, or can he/she recite durood and prayers after the last tashahud? 

3. If a person misses the 4 sunnah prayer prior to Zuhr or Jumah Salaah and joins the farz prayer in jamaat, does he/she needs to offer the missed sunnah prayer afterwards or not? If yes, then what's the correct sequence of offering the missed sunnahs? Should those be offered immediately after the farz jamaat salaah or after the 2 or 4 sunnah prayers that we usually pray after Jumah or Zuhr farz salaah? 

4. If a person makes sajdah while a couple of hair strands cover half the width of his forehead, would the sajdah be valid?

Walking in front of a masbooq

Q: In a small Masjid, the Imam used a sutrah while leading the jamaat (it is a common practice in some places to do second or third jamaat if the main jamaat is missed). Is it permissible to pass in front of people who missed their rakaat with the Imam in these second/third jamaat are completing their prayers. Will they still be considered to be behind sutrah even if Imam has completed his own Prayer?

Imaam moving forward when a second person joins the jamaat

Q: I was leading salaah with one muqtadi, who was standing 1 foot behind me, when a second muqtadi joined the prayer. However, the muqtadis did not move back 1 saff, so I dragged my feet and moved 1 saff forward. However, I dragged my feet 4 times to reach my place.

1. Is the salaah valid?

2. Is it necessary for the imaam to move forward if the muqtadis don't move back?

3. Is it necessary for the muqtadis to move behind?