What is the ruling if someone wipes his head then shaves it - must he repeat it?
If a person performing ablution shaves his head after wiping it during ablution, he is not required to re-wipe his head or repeat the ablution. And Allah the Almighty knows best.
What is the ruling on having an intention (Niyyah) for every prayer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Intention (Niyyah) is a pillar (Rukn) without which the prayer is not valid. The worshiper must have an intention for every prayer, meaning they must consciously intend the act of worship they are performing. Its timing must coincide with the opening Takbir (Takbirat al-Ihram). It is not a requirement to utter it verbally; rather, doing so is considered a recommended Sunnah. There are three levels of intention:
1-If the prayer is obligatory (Fard): It is mandatory to include the Intent (to pray), the Specification (which prayer, e.g., 'Asr), and the Obligation (recognizing it as a Fard). For example, one should bring to mind or say: 'I intend to pray the Fard of 'Asr.'
2-If it is a voluntary prayer restricted by a specific time or cause (Sunnah Muqayyadah): It is mandatory to include the Intent and the Specification. For example: 'I intend to pray the Sunnah before Zuhr' or 'I intend to pray Duha.'
3-If it is an absolute voluntary prayer (Nafl Mutlaq): It is sufficient to simply have the Intent to pray. For example: 'I intend to pray.'
And Allah the Exalted knows best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
Is it valid to perform Tarawih prayer at home?
It is Sunnah for a Muslim to perform Tarawih prayer in congregation at the mosque.
However, if a person sometimes prays it at home in congregation with their family for a valid reason, there is no harm in doing so.