Is it valid to share in the 'aqīqah by contributing a seventh share of a camel or cow?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
Yes, it is permissible. If a group of people jointly share in the slaughter of a camel or a cow on behalf of seven individuals, this is valid — regardless of whether all of them intend the 'aqīqah, or some intend the 'aqīqah, others the uḍḥiyyah, and others simply the purchase of meat. And Allah Almighty knows best.
Is a woman's prayer performed while following an Imam whose prayer is being broadcasted via the television valid, and does she receive the reward of congregational prayer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Among the conditions for the validity of following an Imam (Iqtida’) is that the Imam and the follower must be in the same location. Therefore, the prayer of one who follows an Imam whose prayer is being broadcasted via television from a distant location is not valid. However, if the follower is with the Imam in the same mosque, the prayer is valid. And Allah the Exalted knows best.
What is the ruling on performing dry ablution (tayammum) on a wall or stone?
One of the conditions for tayammum is that it be with pure, clean dust (soil). It is not valid to perform tayammum by striking stones, a wall, sand that has no dust (i.e., that does not produce dust), pebbles alone, or clay, as these are not from the earth or its same category. And Allah the Almighty 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.