Home Interest:
You may be able to claim both the rent and the mortgage interest. In order to claim the mortgage interest, the loan must be in your name and you must itemize deductions on your tax return. You will also be able to deduct the property taxes you paid for the year along with other items.
In case your unfamiliar with what it means to itemize tax deductions, please see the information below.
https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Deductions-and-Credits/Tax-Deduction-Wisdom---Sho...
The rent you paid will be deducted on the state tax return as renter's deduction and the property taxes can usually be claimed here too. The Turbo Tax software automatically carries those taxes over from the federal return. Just keep in mind that there is a limit on how mush of a renter's deduction you can claim. Therefore, you may only be able to deduct just rent or just property taxes on your state return.
Grandchild:
You need to put the amount of time that your grandchild actually lived in your home during the year. You are not required to have had him or her in your home for the for the entire year. However, you must have supported this child in your home for more than 6 months of the year. This is the requirement to claim the child as a dependent and the related benefits: child tax credit, earned income credit, and the child and dependent care credit.
Here is a link you can visit to see about the requirements for these credits:
https://www.eitc.irs.gov/EITC-Central/main
In case your wanting to ensure your eligible to claim your grand child as a dependent, please see the following information.
https://ttlc.intuit.com/replies/4268506
