(2) What is the ruling for tahajjud in such a case is sleeping for a while necessary to pray tahajjud?
بسم الله الرحمن الرحيم
(1) It is not necessary for you to keep on awakening whole night so that you may not miss Fajr salah. Shariah has given some rights to the body as well. It is mentioned in hadith:
ان لجسدك عليك حقا.
Your body has some rights over you.
If, as a result of constant awakening, your health is affected then it shall not be lawful to be awake the whole night.
(2) It is not necessary to offer tahajjud after sleeping for some time. Hence if someone offers salah with the intention of tahajjud after Isha salah it will be acceptable.
Allah knows Best!
Darul Uloom Deoband