Is it obligatory for me to pray all the prescribed prayers in the mosque despite its distance and the difficulty of getting there, and would I be sinful if I prayed Fajr at home?

Praying in the mosque in congregation is one of the best acts of drawing near to Allah, and it is a sign of faith. Allah’s testimony regarding those who tend to His mosques confirms this in His saying: "The mosques of Allah are only tended by those who believe in Allah and the Last Day and establish prayer and give zakat (definition) and do not fear except Allah. For those, perhaps, will be among the rightly guided."

Praying in the mosque also resolves the dispute of those who made it obligatory to pray there, like Ibn Taymiyyah, even though the majority of those who deem congregational prayer obligatory do not require it to be performed in the mosque.

If you are delayed from the congregation due to an excuse, such as sleep, then pray in congregation with your family at home. If you pray in congregation at home without an excuse, you have missed the virtue but have fulfilled the obligation according to the majority of scholars.

According to the view that attending the congregation in the mosque is obligatory, it becomes obligatory upon you if you can hear the call (adhan (definition)) in normal circumstances without a loudspeaker; otherwise, it is not obligatory for you, as mentioned by Ibn Baz.

So, if you can hear the normal call without a loudspeaker when sounds are quiet, then praying in the mosque is obligatory for you. Otherwise, it is permissible for you to pray at home or with neighbors, based on the Prophet's (peace be upon him) saying to the blind man: "Do you hear the call to prayer?" He said, "Yes." He said, "Then respond." And his saying: "Whoever hears the call and does not come, there is no prayer for him except for an excuse." And walking to the mosque earns you reward for both going and coming back.