Is keeping money in and taking profits from the Islamic Bank lawful? 

If Allah Wills this is lawful, because they stipulate on theselves to work in accordance to the guidlines of Islamic Law. And Alalh Knows Best.

What is the ruling on giving my mother a gift at "Mother's day", is it Halal or Haram?

Praise be to Allah the Lord of the Worlds.

There is nothing wrong with doing that as long as it isn`t done in imitation of the disbelievers. And Allah The Almighty Knows Best.

Is it permissible to offer prayer at home, or should it be offered in the mosque?

All perfect praise be to Allah, The Lord of The Worlds.                                                                                                                                                                      It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.

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 .