Is it permissible for a person who had vowed to give a certain amount of money to another, but didn`t find the latter to donate that money to the mosque?
In principle, the vowing person should abide by his vow as much as possible. Therefore, if he couldn`t find the person that he had made the vow for, then the vow itself is countless and nothing is due on its maker. However, if the latter happens to find the former later on then, he has to give him that money.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
I am a wholesale gold trader. My clients are retail gold shop owners. I sell them gold jewelry and ornaments on credit, and they repay me in weekly installments over a period of two months or more. The payments are made in cash for the labor charges and for the gold they owe me. In return, they may give me either cash or used gold.
It is not permissible to sell gold or silver on an installment basis or with a deferred payment; rather, immediate exchange is required. The Messenger of Allah (PBUH) said: "Gold for gold, silver for silver, wheat for wheat, barley for barley, dates for dates, and salt for salt, like for like, hand to hand. Whoever gives more or asks for more has engaged in usury; the giver and receiver are the same in this regard." (Muslim). Additionally, the weight of the gold must be equal when exchanging new for used; otherwise, it involves usury. The solution is to buy the used gold with cash, then sell the new gold for cash as well. However, payment must be made at the time of the transaction in both cases. Alternatively, one could take the used gold with the intention of refining it, reshape it into a new form, or repair it, then take a fee for the workmanship or repair. And Allah Knows Best
Is it permissible to give Zakah (obligatory charity) to one`s poor sister, or brother?
It is permissible for one to give the Zakah to his poor sister, or brother if providing for them isn`t due on him, and the sister doesn`t receive sufficient provision from whomever that is due on. And Allah Knows Best.