Is it obligatory to have an intention (niyyah) for every prayer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is obligatory for the worshiper to have a specific intention (Niyyah) for every prayer, meaning they must consciously intend the act of worship they are performing. The intention is a pillar (Rukn) of the prayer, and the prayer is not valid without it. It is not a requirement to utter the intention verbally; rather, doing so is considered a recommended Sunnah. And Allah the Exalted knows best.
Does the time of Fajr (dawn) prayer start after the first Athaan (call for prayer), or the second one?
All perfect praise be to Allah,The Lord of The Worlds The time of Fajr prayer starts after the second Adhan. And Allah Knows Best.
What should one who doubts the validity of his ablution and performs it frequently do?
Whoever doubts the validity of his ablution after finishing it, his ablution is valid. This is because the default is the validity of the ablution, and because doubt after finishing an act of worship doesn`t affect its validity. And Allah the Almighty knows best.
What is the ruling on performing voluntary (nafl) prayer during the time of istiwā' (the sun's zenith) on the day of Jumu‘ah (Friday), and is this ruling specific to someone who has come to the mosque to perform the Jumu‘ah prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is permissible to perform voluntary (nafl) prayer during the time of istiwā' (the sun's zenith, i.e., its highest point before noon) on the day of Jumu‘ah (Friday) — whether for someone who is attending the Friday prayer or someone who is not, and whether they are among the people obligated to attend it or not.
It is stated in Mughnī al-Muḥtāj (Vol.1/P.310): "The more correct position (al-aṣaḥḥ) is that prayer is permissible during this time without restriction — regardless of whether the person will be attending the Jumu‘ah prayer or not." And Allah, the Exalted, knows best.