The undisputed fact is that your mother is owner of half share of the property left by your late father. As far as answers to your queries, the same are as follows
1) Your mother can sell her undivided share of property and the buyer shall acquire interest in the property as a co-owner but if he is outside the family he can not enjoy the joint possession and has to go for partition of property. The tenant might be interest to buy the property since he is in possession of the same and it will be upon you to repossess the property through a process of law. the Registrar shall have no objection in adding the buyer's name as co-owner in the title of the property. If you can understand the law I would like to produce section 44 of the Transfer of Property Act which is as follows:
Transfer by one co-owner.—Where one of two or more co-owners of immoveable property legally competent in that behalf transfers his share of such property or any interest therein, the transferee acquires as to such share or interest, and so far as is necessary to give, effect to the transfer, the transferor’s right to joint possession or other common or part enjoyment of the property, and to enforce a partition of the same, but subject to the conditions and liabilities affecting at the date of the transfer, the share or interest so transferred. Where the transferee of a share of a dwelling-house belonging to an undivided family is not a member of the family, nothing in this section shall be deemed to entitle him to joint possession or other common or part enjoyment of the house.
2) Can not gift the property without your consent. Any such gift deed would be void.
3) She can Will her property as per her desires but it is open for you to challenge the will on the basis that the beneficiary had control over the executrix and was influenced by him.
What you should do in the circumstance is that you file a suit for injunction, restraining your mother from disposing her property by way of sale to the tenant as the buyer being in possession shall possess the property jointly thus violating the law. Implead the tenant as a party. Also restrain her from executing any gift deed in favour of a third party without your consent.