1. If the contents of the family tree document are correct then there is no legal infirmity about this document.
2. No it is not necessary
3. It is an authenticated documents containing the correct information, hence if at all the mother has any grievance about the correctness of the facts stated therein then the mother can record her objection to this, otherwise there is no such representation maintainable especially for the reasons you have stated.
4. She can claim a share out of her son's share in that property.
5. If the spouse has been nominated in that account, she has to share the balance with the other class i legal heir i.e., the mother of the deceased account holder.
6. No, they have been intentionally done by your brother during his lifetime, hence your mother cannot claim any share out of that transferred amount.
7. Your mother has to issue a legal notice asking for her share and file a money recovery suit in the court.
8. She may either have to compensate with money to the value equal to her share or she may sell the same and consideration amount may be equally distributed between both.
9. She has find them out through her own sources and claim hr rightful share in that also.
10. If the legal heirs are inheriting the assets of the deceased they are equally liable to settle the debts incurred or remaining outstanding in the name of the deceased.
11. They both have equal rights in the property, they can partition either amicably or through court of law.
12. No, however if your father is reported to die intestate then she may get a share out of her deceased husband's share in it as his legal heir.