What is the ruling on using bank cards such as Visa and MasterCard?

All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upo all of his family and comapnions.

It is permissible to use such cards in accordance with the following conditions : 1- The issuing body does not stipulate collecting interest upon late payments. 2- The merchant does not increase the price of the goods due to an increase in the commission collected from him by the card issuer.3- The purchased item(s) should not be gold , silver or currency in case the card is not covered. And Allah Knows Best.

Is it permissible for a woman who broke fasting due to delivery to pay a ransom ?

She should make up for the missed fasting days once she becomes ritually pure, but it is impermissible for her to pay a ransom while being able to fast.

Should one fulfill his vow of donating an animal to the poor and needy with one sheep although he had repeated the vow several times?

Each vow has to be fulfilled, and repeating the words of the same vow for confirmation renders its fulfillment obligatory as one vow. As for multiple vows, they have to be fulfilled as well.

Is it permissible for a woman to swear a false oath by the Holy Quran to please her distrustful husband?

It is forbidden for a person to do so since this is considered a false oath(Yamaan Ghamoos) which dips(Yaghmos) its taker in Hellfire if he/she doesn`t seek repentance, and he/she has to expiate for that oath. You should avoid any doubtful acts so that your husband`s chest remains clear from the evil suggestions of the devil.