Your question has several facets to it. I will try to answer them individually.
Is your mother's SSA included as income for the Gross Income calculation? Yes and no. Social Security can become taxable income. However, if your mother's SSA were taxable, you would not be able to claim her as a dependent. So TurboTax is right that the Social Security cannot be counted for Gross Income calculation. However, Social Security does count towards the Support Calculation. To claim your mother as a dependent, she must have made less than $4050 of income and you must provide at least half of her support. You might not be able to claim her as a dependent because of what you state in the question (paying $700 of $4000 monthly expenses isn't half of her support).
Can you claim those expenses? Probably as a medical expense. Although your mother may not count as a dependent, you are still allowed to count these medical expenses if the only reason you can't claim your mother is because of the support calculation above. You can claim them because you paid them. Medical expenses are claimed on Schedule A as part of your itemized deductions. Please see this FAQ with more information: https://ttlc.intuit.com/replies/3302335
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