1. Yes, the mother or any of the legal heirs can bequeath her/his share of the property in favour of any body,
2. Yes, the mother or any of the legal heirs can gift/relinquish/settle her/his share of the property in favour of any body,
3. If he refuses to vacate, you shall have to lodge a police complaint and also file an eviction suit against the unauthorised occupier,
4. If rent agreement is executed, then the occupier will have the legal right to stay therein by virtue of the said rent agreement as per local Tenancy Act. Do not make this mistake. Rather make it mentioned in the settlement deed that all will vacate the particular house in favour of its settler and will occupy their own settled property only.