Is it a condition that the meat of the Eid sacrifice should be distributed in order for the sacrifice to be accepted by Allah?

The poor and needy should be given from that sacrifice , even if it be a small portion,(One kilo for example) as raw meat so that they would cook it the way they want. The remainder of that sacrifice is either distributed, eaten, or offered to people in a banquette .

Who is required to fast?

Any adult, sane, and hardship enduring Muslim person is required to fast unless there were prohibiting conditions such as menstruation and confinement.

Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings involve acts of sin?

Maintaining kinship ties is mandatory, thus if visiting your blood-relatives hinders their sinful acts , then you should do so. However, if their sinful acts persist while you are at their gathering, then it is sufficient that you maintain kinship ties via telephone and the like.

I`m a cashier at a branch of the Civil Service Consumer Corporation. Sometimes, customers pay more than the due amount. Do I have the right to take that extra money knowing that I cover shortages from my own salary?