Your friend should report the rent you pay her as rent income and deduct the rent she pays as rent expense. Even though the two amounts cancel each other out so that there is no taxable income, that would be the correct way to handle it from a tax perspective. You don't need to issue any tax documents reporting the rent since it is not a commercial transaction.
The tax treatment would be the same no matter how the rent is paid, it just may attract less attention by the IRS if you made the payment directly to the landlord. If your name was added to the lease that would mean you were renting from the landlord, so that would mean your friend wouldn't have to worry about reporting the rent on her tax return.
To deduct home office expenses, you would need to be self-employed, as the home office deduction is not available for salary employees. You could do that if you were self-employed and used the office space exclusively for business purposes and you didn't have an office elsewhere you used for the business. The home office would have to be necessary for business purposes.
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