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Normally you must be unmarried to claim head of household. You can be married and still claim if you were still legally married as of December 31, you are considered unmarried (and therefore eligible for Head of Household) if all 5 of these conditions apply:
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Yes you have to file jointly or you will be taxed on you SS benefits. If you only receive SSDI benefits then you should not be taxed on those benefits but if you have other income then you might. Here are the rules from SSA on taxation of SS benefits: https://www.ssa.gov/planners/taxes.html
"No one pays federal income tax on more than 85 percent of his or her Social Security benefits based on Internal Revenue Service (IRS) rules. If you:
more than $34,000, up to 85 percent of your benefits may be taxable.
more than $44,000, up to 85 percent of your benefits may be taxable.