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In whose name property stands.  Your mother's sisters can claim their share in that property because after the death of the father succession opens.  Whether your grandfather has made any arrangements like will or gift in your mother's name? if so if it is self acquired property of GF then your mother's sisters cannot claim anything.

When u say house rent bill. That means ur grandfather was a tenant. Any one who continues to live in tenantated house becomes a tenant and has to ask for name change in rent receipt. Those who do not live in the tenanted premises loose the right.

Preserve the documents which r in ur mothers favour. Write to landlord thru a good rent act advocate and get the rent receipt transfeered in her name.Non residing heirs have no right over tenanted premises. As property prices have gone up and redevlopments talk r everywhere those who were not interested earlier r now suddenly claimimg rights. Wasteful efforts.

1) Can they put a case on her and ask her to leave?

No. She may claim her share of that property if the property is ancestral property. She may not claim her share if it is not ancestral property like your grandfather self acquired property but the self acquired property would have been transferred to your mother name throw gift or will deed otherwise your mother sister will get her share.

 2) Is my mom entitled to get her share?

Yes. Your mother definitely will get her share.

 3) are we liable to pay them?

No. You need not pay any single amount.