1. Your mom can definitely travel to the USA. But that should be intermittent. If she shifts there permanently then the landlord can claim that she is not using the property for which it was let to the tenant and therefore is liable to be evicted. So intermittently she can travel to the USA to reside with her children and to visit them. But she cannot keep the house locked for long. If the house is locked for more than 6 months then the landlord can have a claim of eviction against her that she no more needs the premises. 

2. After your mother the tenancy of the house will go to her heirs. But the same restriction as above will also apply to her heirs ie you and your sister 

On father demise your mother ,you and sister would be legal heirs 

2) your mother can visit you for some months but she has to stay in said premises during the time she is in Bombay 

3) you cannot lock the house after mother demise 

4) you can with landlord consent sell your tenancy rights 

You have to stay in the house. The property passes on to the legal heirs. 

1. Your mother can get the rental agreement transferred to her name after the death of your father, however if she is not residing there in the rented property for a long period then the landlord may issue an eviction notice to vacate the property as it is not occupied for a long period. 

2. The rented property will not devolve on the legal heirs automatically upon the death of the original tenant.

The rights of the legal heirs will be conferred only when they were living with the tenant in the rented premises at the time of death, hence you my have to be careful about it.

The landlord may take possession of the property if it is not occupied for a long period.

The house under lock and key is not a permission for you to retain the pagdi system property .

There are no inheritance rights in the rented property to the legal heirs of the deceased tenant.

1. Yes, as per law she will have to stay in the property as Pagdi ownership can be claimed by the legal heirs who are residing in the property. 

2. The pagdi system was legalized by the Maharashtra Rent Control Act, 1999, and upon the death of a pagdi tenant, the tenancy rights are transferred to the legal heir/s who were living in the property when the tenant died.

She needs to stay there before or after on sometime earlier. Without stating there for any period she may have issues for claiming tenancy

Dear Sir

Your mother will be required to live on the property most of the time, but occasionally travelling to the US to visit you for a short period will not be a problem.

After your mother's time, you and your sister will be the legal heirs. However if you lock up the house and not stay there, the landlord has the right to evict you as tenants.

Thank you