Hi

1) A property will go the entire legal heirs of the deceased and the B property's half share will go the entire legal heirs while the the other half will remain with the co owner;

2) A society can transfer the A property to the nominee provided there is no dispute or rival claim, in case of a rival claim, the society may call for a succession certificate.

3)  Mere waste of time to discuss on this issue because this is a civil matter and dispute also is of civil nature;

4) and 5) The answer is given in 2) above.

6)  The subsistence of a marriage of the second wife with some body at the time of marrying your father has to be proved with substantial and authentic evidences, if proved, she stands losing her claim as a legal heir for the fact that her subsequent marriage with your father while her subsisting marriage with some one was not divorced as on that date, her subsequent marriage becomes null and void hence she cannot claim her status as wife of your father thus lost all her rights as a legal heir to the deceased, however, her son has a right over the property as a legal heir of your father, with the status as a son born through a concubine.

Mr. Kalaiselvan is rightly advise