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.
What is the ruling on using a patch to suppress hunger or a nicotine patch while fasting?
Fasting is a great act of worship and one of the pillars of Islam. If people knew the immense reward of Ramadan, they would wish for the whole year to be Ramadan.
Whoever eats Suhoor and breaks their fast according to the Sunnah will not experience extreme hardship, making such patches unnecessary.
However, using these patches does not break the fast because they are not a source of nourishment and do not enter the body cavity (jauf) through an open passage.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Is it permissible to pay the expiation of an oath to a charitable party as it feeds the poor and needy such as Tkiyet Um Ali?
There is no harm on authorizing someone to handle the expiation of another to the poor and needy, since Shafiee's jurists stated that authorizing someone for the purpose of distributing Zakah, expiation and vow is permissible as stated in [Mughni Al-Muhtajj,vol.3/pp.237]. At last, the intention of the person who intends to authorize someone else suffices in this regard. And Allah Knows Best.