Witr and Nawaafil

Can one perform the Sunnats of fajr after the Fardh?

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?

Tahajjud Salaah

Q: I want to know is it compulsory to sleep before performing tahujjud Salaah? If a person is awake till 3 am, can he/she then perform the Salaah? I have heard from many people that it's necessary to sleep before.

Witr Salaah

Q: My question is regarding witr Salaah. In one Hadith the prophet (Sallallahu Alayhi Wasallam) forbade praying witr like maghrib but in the hanafi fiqh it is prayed like maghrib and I am a bit confused. Please clarify.