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 us to sever ties of kinship if our blood-relatives` gatherings/meetings involve acts of sin?

Observing kinship ties is obligatory, and paying your blood-relatives a visit causing their sinful acts to be hindered , then you should do so. However, if their sinful acts continue while you are at their gathering and they didn't respond positively, then observing kinship ties via telephone and the like will suffice in this case. And Allah Knows 

Is Zakah (obligatory charity) due on articles of merchandise even if they were stacked in the merchant`s stores for years?

Yes, Zakah is due on goods even if they were stacked in merchant`s stores for years, in this manner Islam struggle against monopoly.

I am a doctor, and sometimes I refer patients to the hospital for surgical interventions, and they give me a commission from the fees of the surgeries, even though I do not perform these surgeries. What is the ruling?