Do I have to claim the $1600 my mom gives me a month as income?

Unless you have a formal caregiver contract set up legally this probably falls more into the category of a gift than income, which can have serious repercussions if she needs to apply for medicaid within the next 5 years. I think it is time you and your mother stop playing this silly game and she just pays her own bills.

Nope, just save the receipts from the rent you pay.