Fajr

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.

Yes. Every Nafilah Prayer is usually two Rak'ats like Fajr Prayer, but Nafilah Prayer can be performed with reciting Sura Al-Hamd only with out other Sura, although it is better to recite another Sura in Nafilah, while it is obligatory in Fajr Prayers to recite another Sura after Sura Al-Hamd.

Nafilah Prayer can be recited while sitting with out illness and even while walking intentionally, and when it is recited while walking, no need to face Qibla. There are other facilities in Nafilah Prayers. 

Wassalam.

Yes, it is Makrouh (disliked and harmful) to sleep between Fajr and sunrise and also between Asr and Maghrib.

Sleeping between Fajr and sunrise drives away Barakat and makes the income less, as we read in many Hadeeths.(النوم بين الفجر وطلوع الشمس يطرد الرزق)

Sleeping between Asr and Maghrib harms the mental and psychological health. The Hadeeth states: Who sleeps after Asr,  should not blame but himself. (من نام بعد العصر فلا يلومن الا نفسه)

Wassalam.

It is allowed but Makrouh (Disliked) as we have a Hadeeth from Imam Mohammad Al-Baqir (AS) stating that it is Makrouh to have sexual intercourse between Fajr and sunrise.

Wasa'il Al-Shia by Al-Hurr Al-Aamili, Vol 20, Page 125.

عن عبد الرحمن بن سالم، عن أبيه، عن أبي جعفر (عليه السلام) قال: قلت له: هل يكره الجماع في وقت من الأوقات وإن كان حلالا؟ قال:
نعم ما بين طلوع الفجر إلى طلوع الشمس، ومن مغيب الشمس إلى مغيب الشفق، وفى اليوم الذي تنكسف فيه    الشمس، وفى الليلة التي ينكسف فيها

القمر.

The Hadeeths explain that it is recommended to be awake between Fajr and sunrise, by being busy in worshiping Allah. Sexual relationship with spouse is recommended in general but there are certain times and situations where it becomes Makrouh.

Wassalam