There is no difference between Fajr prayer and subh prayer. They are the same. Fajr prayer consists of two rak’ahs. The time for Fajr prayer starts when the true dawn begins, and lasts until the sun rises.

It is not allowed for a woman to call the Adhan, according to all the reports. Going against this is going against the way of the believers.

Sujud as-Sahw in prayer is prescribed for three reasons: 1- Doing something extra, 2- Omitting something, 3- Being uncertain. Sujud as-sawh should be done before the salam if you have omitted one of the obligatory parts or you are not sure how many rak`ahs you have done. Sujud as-sahw should be done after the salam if you have added something extra or you are not certain but one of the two choices seems more likely to be the case.

To get rid of bad thoughts about Allah, you should say, “Amantu Billahi wa Rasulihi (I believe in Allah and His Messenger).” Try to stop thinking about that as much as possible, and keep busy with things that will distract you from it.

It is forbidden to pray after Fajr prayer until the sun has risen above the horizon to the height of a spear, at the time of noon when the sun is at its zenith, and after the time of `Asr until the sun is fully set.

Qunut is the name of a du’a (supplication) offered during prayer at a specific point while standing. Qunut is recommended in witr prayer and when a calamity befalls Muslims. For more, see the detailed answer.