If you have no other income besides social security, you would not need to file a tax return. If you did, you would have no taxable income to claim any credits and filing as Head of Household would not give you any type of refund. In order to get any type of refund of money you did not pay in, you would have to have earned income.
No, you could not claim her as a dependent if she is getting alimony of about $6,000 per year as she would be over the $4,700 threshold to claim a dependent.
That's complicated.
First, the expert is only partly right about the alimony. You can't claim an adult as a dependent if they have more than $4700 of gross taxable income. (From the IRS instructions, "Gross income is all income in the form of money, property, and services that isn't exempt from tax.") Alimony may or may not be taxable, depending on when the divorce was finalized.
To claim your sister as a dependent, she must have taxable income less than $4700 AND you pay more than half her living expenses. This does not mean you have more money than she does, it depends on how you use that money. For example, if your sister donates all her money to charity and you pay her expenses, then you can claim her, even though she might have more income than you. But you have to add up her personal expenses and determine if you paid more than half. There is a worksheet in publication 501.
Then, for HOH, you must also pay more than half the total household expenses. Again, this is about who pays, not just who has more money. However, if her incoming funds are more than yours (because she has alimony and SS and you only have SS), it may be tricky to show that you pay more than half the total housing expenses.
And lastly, if you have no taxable income, there is no benefit from claiming a dependent (no free money).