Fajr

Yes, she has to perform Qadha for Morning Prayer which she missed after being clear from the menstruation. She should have performed Tayammum and prayed on the time of Morning Prayer if Ghusl was not possible.

Wassalam.

It is not allowed for a Muslim to deliberately delay his Prayer till its time goes away. Muslim must do every thing possible to wake up and perform Fajr Prayer withing its time.

If he is ill and could not get up for the Prayer, he must perform it as Qadha later on. 

Wassalam.

Tahajjud time is during the night ( starting from either midnight or even after Isha) and ending with Fajr time. Tahajjud can be also performed after Fajr as Qadha.

Wassalam.

The real time of Fajr is when the light starts appearing on the horizon. This is called true dawn or Al- Fajr al-Sadiq. 
This light on the horizon increases gradually. We call it true dawn to distinguish between it and false dawn or zodiac dawn which happens before the true dawn.
False dawn looks like a mostly vertical column or pyramid, and it keeps on decreasing, in contrast to the horizontal increasing true dawn.

The real dawn is the time of morning prayer and end of Suhoor. It is good to stop eating and drinking little while before that to be sure that you did not eat or drink after Fajr.

If you know the exact time of real Fajr, you can eat and drink till a moment before it, but many people stop eating and drinking about ten minutes before the published time of Fajr because they are not sure about the exact time of Fajr, so they want to do precaution.

Wassalam.

It is compulsory on every one of them to perform Ghusl of Janabah latest before Fajr (Dawn) time. Remaining deliberately on the state of Janabah will invalidate the fast and cause a major sin with all its sequences which are Qadha of the fast of that day and Kaffara of feeding sixty poor persons or fasting two months. Ghusl must be performed before the time of Fajr.

Wassalam.

It is compulsory to pray Fajr prayers and every prayer within it’s
time before it lapses (i.e. becomes Qadha). The Mo’min must do
everything that he can to perform all his prayers within its time.

You need to change your sleeping timing to be sure that you can wake up
to perform the Fajr prayer. The Hadith says : He who sleeps during the
time of Fajr prayer and misses the prayer will be called from the sky
: O loser !

If for any reason like illness etc. You could not wake up for Fajr,
then you must perform it as Qadha on the same day or better as soon as
you wake but that can never be usual procedure. You need to set an
alarm or request your family members to wake you up or your friends to
phone you to be sure that you wake up for Fajr prayers on time.
According to authentic Hadeeths, the loss of a person missing Fajr
prayer is more than all the treasures of this world.

Wassalam.