Is it valid to perform the obligatory prayer while sitting?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Standing (Al-Qiyam) while having the ability to do so is one of the pillars (Arkan) of the prayer; the obligatory (Faridah) prayer is not valid without it. However, whoever is unable to stand may pray sitting down. As for voluntary (Nafilah) prayers, it is permissible to pray them sitting even if one is able to stand, but the one sitting receives half the reward of the one who prays standing. And Allah the Exalted knows best.
What is the ruling on entering the toilet with something containing the remembrance of Allah?
It is disliked to bring into the toilet anything containing a reminder of Allah, such as the Quran or a sacred name (like the name of Allah, Muhammad, etc.). If one does that, etiquette dictates that he conceal it so it is not visible while entering the toilet, either by cupping his hands over it, putting it in his pocket, or inside his shirt if it is a necklace or ring. And Allah the Almighty knows best.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
What is the ruling on reciting aloud (*jahr*) in a silent (*sirriyyah*) prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Reciting aloud (jahr) in the Fajr prayer, in the first two rakʿahs of Maghrib and ʿIshāʾ, and reciting silently (isrār) in the Dhuhr and ʿAṣr prayers is a sunnah. If the worshipper recites aloud where silence is prescribed, or the reverse, his prayer is valid, although it is better to follow the sunnah, in imitation of the Prophet (peace and blessings be upon him). Abū Hurayrah (may Allah be pleased with him) said: “In every prayer there is recitation. What the Messenger of Allah (peace and blessings be upon him) let us hear, we let you hear, and what he concealed from us, we conceal from you. If you add nothing to Umm al-Qurʾān (al-Fātiḥah), it suffices, and if you add to it, it is better.” Narrated by al-Bukhārī.
Shaykh al-Islām Imam al-Nawawī (may Allah have mercy on him) said in al-Majmūʿ (Vol.3/389): “The sunnah is to recite aloud in the two rakʿahs of Ṣubḥ (Fajr), Maghrib, and ʿIshāʾ, and in the Friday prayer, and to recite silently in Dhuhr, ʿAṣr, the third rakʿah of Maghrib, and the third and fourth of ʿIshāʾ. All of this is by the consensus of the Muslims, together with the mutually reinforcing authentic ḥadīths on the matter. This is the ruling for the imam. As for the individual worshipper (munfarid), it is sunnah for him to recite aloud according to us and according to the majority… If he recites aloud where silence is prescribed, or the reverse, his prayer is not invalidated, nor is there a prostration of forgetfulness (sujūd al-sahw) for it, but he has committed something disliked (makrūh).” And Allah Almighty knows best.