sunnats of fajr

Q: As we know that in fajar namaz if we are late and in danger of missing the farz prayer then we do not offer the 2 rakat sunnah prayer and directly offcer the farz prayer snd join the imam as muqtadi. I want to know that do we have to offer the 2 rakat sunnah prayer after the farz prayer? Kindly guide me. I am talking about fajar prayer.

Q: I want to know that when I enter the masjid and the imam is in the 2nd rakat of the fajr prayer, should I first offer my 2 rakat sunnah or directly join the imam in the 2nd rakat of the fajr prayer?

Q: When I woke up for fajar and after I performed wudhu there were 3 minutes left for sunrise. I therefore performed only the 2 rakahs fardh fajar and did not offer the 2 rakah sunnah. Was I correct?

Should I offer the 2 rakahs sunnah later?

Q: I overslept and was late awakening for the fajr Salaah. Thinking it to be very late, I prayed wudhu and only the two fard of Fajr. However after reading this, I checked the perpetual calendar and realised there was still 11 minutes before sunrise. I immediately prayed both my sunnah and my fard of fajr, with 3 minutes still left for sunrise. Was this correct of me? What should I have done in this situation. Is my salaah valid?

Q: I woke up for fajr and thought there was not enough time to read the sunnahs, so I read the fard. After completing the fard, I realized there was still time left, so I read the sunnahs of Fajr after praying the fard of Fajr, within the time of Fajr. are the sunnahs valid? Or will I need to redo them after sunrise?