A. After the demise of your father without a Will, all class I legal heirs are the absolute owner of the property. Hence, your mother can claim a share over the 50% share in the property.
B. If you live in a Rented House you are eligible for something called House Rent Allowance and this has tax benefits.
C. Payment of Rent to Parents for House Rent Allowance: This is a tricky question. Legally there is nothing stopping you from doing this but this is a concept that is frowned upon because effectively this amounts to cheating. In real life - would we really pay rent to our mom or dad? this is crucial issue.
Anyways, if you insist on going this route, you need to understand that if your tax returns get audited in future, you would have to answer the tax officials on why you claimed as if you are paying rent to your own parents plus prove that you actually paid rent. Otherwise - the penalties you will pay and the consequences would be much worse than the few thousand rupees of tax you will save.
If you wish to go this route - as a first step, you would need to apply for a PAN Card for your mom (on whoever's name you want to pay rent). Only the person on whose name the house is registered can receive rent. If you want to show rent to your mom who is a housewife - that is not allowed. After this you would have to sign a rental agreement on stamp paper and register the same for the amount of rent that is nominal for your area plus the size of your respective share. Once done, you must make sure to formally transfer the rent to your mom account plus ask them to issue a rent receipt so that you can use the same for tax filing purposes. (Sources Mr. Anand Vijayakumar blogspot).