What is the ruling on eating or drinking during the first Adhan?
It is permissible to eat and drink during the first Adhan because it takes place before the break of dawn. The purpose of this Adhan is to notify Muslims of the approaching dawn so they can prepare to stop eating when they hear the second Adhan.
Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.
What is the ruling of Islamic Law on kissing while fasting?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is prohibitively disliked (makrūh taḥrīman) for a husband to engage in foreplay with or kiss his wife during the daylight hours of Ramaḍān if doing so stirs his desire. If pre-seminal fluid (madhī) is released as a result, the fast is not invalidated; however, if seminal fluid (manī) is released, the fast is broken.
Imām al-Khaṭīb al-Sharbīnī, may Allah have mercy upon him, stated: "It is prohibitively disliked to kiss — whether on the mouth or elsewhere — for one whose desire is stirred, whether man or woman, to the extent that he fears it may lead to intercourse or ejaculation. Embracing, touching, and similar acts without a barrier carry the same ruling as kissing in this regard, because such acts expose the act of worship to being corrupted — and as the two Ṣaḥīḥs record: 'Whoever circles around a protected boundary is liable to fall into it.'"
He further added: "It is preferable for one whose desire is not stirred — even if he is a young man — to refrain from such acts as well, so as to close the door entirely. For he may believe his desire has not been aroused when in fact it has. Moreover, it is Sunnah for the fasting person to abstain from all desires without exception." — Summarised with minor adaptation from [Mughnī al-Muḥtāj]. And Allah the Almighty 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?!