What is the ruling on seeking forgiveness between the two Friday sermons?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to seek forgiveness (Istighfar) between the two Friday sermons (Khutbatayn), and it does not invalidate the Friday prayer (Salat al-Jumu'ah). Furthermore, it is not disliked (Makruh) to speak before the sermon, after it, or between the two sermons. Likewise, it is not disliked for one who enters during the sermon to speak if there is a need for it, provided he has not yet taken a place and settled into it. And Allah the Exalted knows best.
What do you say about someone who stays up all night on Laylat al-Qadr but misses Fajr prayer?
Such a person has deprived themselves of a great reward. Performing obligatory acts is more beloved to Allah than voluntary acts. Fajr prayer is an obligation, and performing it in congregation is equivalent to worshiping for the entire night. So how can someone neglect it while striving for voluntary prayers?!
Should missed Sunnah prayer be made up?
Making up missed voluntary acts of worship is from Sunnah.
What is the ruling on the ablution of one who washes his arms from the wrist to the elbows?
In the school of Imam al-Shafi'i (may Allah have mercy on him), washing the arms (hands) is achieved by washing the arms completely, from the fingertips to the elbows. Washing only the palms at the beginning is insufficient, as washing them at the beginning is a Sunnah, but after washing the face, it becomes obligatory (fard). The person performing ablution must wash his palms along with the arms after washing the face. If he does not wash his palms, his ablution is invalid and not correct. And Allah the Almighty knows best.