الجواب وبالله التوفيق
1- The time for fajr salah starts at the beginning of Subh-Sadiq(true dawn) and ends at the sunrise.
2- The method you have described to perform sajda sahw in your question is correct according to maslak al-Hanafi, according to maslak al-Shaf‘ai sajda sahw is before performing salam. By reading additional (or less number of) tasbih, sajda-sahw does not become wajib, however the number of sujud should be two.
3- C is the correct tasbih.
4- Yes, if one has wudhu already, during performing a fresh wudhu one can perform masah over the socks.
5- Yes, you can perform those actions as mentioned in your question.
واللہ اعلم بالصواب