Is it permissible for women to gather in one of their homes to pray Tarawih?
Yes, it is recommended (mustahabb) for women to pray Tarawih in congregation if they gather, even if it is not in the mosque.
In fact, praying at home is better for them, as it avoids the potential concerns associated with going to the mosque.
What is the ruling on bypassing a nearby mosque for a distant one?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
If a distant mosque has a larger congregation (Jama‘ah) while the congregation in the nearby mosque is smaller, and the congregation in the nearby mosque is not adversely affected by the worshiper's absence, then it is better for him to pray in the distant mosque with the larger congregation. However, if the congregation in the nearby mosque would be affected by his absence—such as if he is its Imam, or if his presence encourages others to attend—then praying in the nearby mosque is better. This ensures that the congregation is established in two different locations within the community. This is based on the saying of the Prophet ﷺ: 'A man's prayer offered with another man is purer than his prayer which he offers alone, and his prayer with two men is purer than his prayer with one and if they are more (in number), it is more beloved to Allah, the Mighty the Majestic' (Narrated by Ahmad, Abu Dawood, and An-Nasa'i). And Allah the Exalted knows best.
Is it permissible for the guardian to give Sadaqa (voluntary charity) from the money of the orphans?
It is impermissible for the guardian to donate from the money of the orphans because he is entrusted with the safekeeping of that money, and is prohibited from donating it.
What is the ruling on swearing on the Holy Quran and breaking that oath?
It is impermissible to swear on the Holy Quran in vain, and it is also impermissible to subject it to such an act since breaking the oath after swearing on the Quran entails an expiation.