"1. Does funding HSA post tax lower my taxable income (similar to how a Roth IRA might work)? "
An HSA reduces your taxable income. A Roth IRA does not reduce your taxable income. You may be thinking of a traditional pre-tax (deductible) IRA. If you make deductible IRA contributions to a traditional pre-tax IRA, that reduces your taxable income, and you pay the tax when you withdraw the funds. Contributions to a Roth IRA are not deductible and do not reduce your taxable income, so you pay the tax now but you don't pay tax when you withdraw the funds.
Also, if your employer allows you to make pre-tax HSA contributions through payroll deduction, that is better than contributing your after tax dollars. With after-tax contributions, your state and federal income tax is reduced. With payroll contributions, your social security and medicare withholding is also reduced, so you save an extra 7.65%. Even if your current employer is not sponsoring the HSA, you do get the state and federal deduction if you make after-tax contributions.
Also, if your spouse is covered by your family HDHP, then she is eligible to contribute to an HSA in her own name. Your family maximum is the same ($7750 for 2023, or less, depending on when you enrolled in the qualifying HDHP), but you could split it between the two accounts. An HSA is owned by only one person, there are no joint HSAs even if you are married, and there may be reasons to have an account in each spouse's name.
"2. Does this all check out?"
More or less. The Roth idea doesn't help, see above. You can reduce your taxable income by contributing to a pre-tax (deductible) IRA, if your income is within the right limits, but you can't take the money out again until you retire or you pay income tax plus a 10% penalty. It's good for the future but it's not the same as "washing" money through the HSA, where you deposit money, get the deduction, then withdraw it immediately to pay current bills. Like the HSA, your spouse can contribute to an IRA in her own name, even if she does not work, by using your income as the basis for her contributions.
If you have a schedule C self-employment side gig of some kind, you may also qualify for a self -employed 401k plan. This has much higher contribution limits that an IRA (up to $66,000, or up to your self-employment income, instead of a maximum of $6500). Again however, while this is a substantial tax deduction, it locks the money away until retirement, and doesn't give you the ability to take a tax deduction and still spend the money, like the HSA does.
https://www.irs.gov/retirement-plans/retirement-plans-for-self-employed-people
Yes, you can also "wash" money through a 529 plan to pay your student loan, and you will get a state tax deduction. Any interest or investment gains will be tax-free, but you may not have much in the way of gains the way you plan to use the account. You may be allowed to contribute much more than $10,000, depending on your state, but you can only withdraw $10,000 for student loan payments. So any contributions over that amount would have to be used for future student loan payments or your child's expenses. (You must change the beneficiary of the account from yourself to your child after you are finished with the student loan.)
Depending on your state income tax rate, you might save $500-$1000 in state income taxes this way. However, I would not completely discount the federal student loan deduction. Even though interest may not be accruing, I suspect you are still paying the interest that accrued before the pandemic. Check with your lender. (It would be extraordinarily generous if they applied 100% of your payments to the principal.) If you did have $2500 in interest payments, that could save you $550 in federal tax and $125-$250 in state tax, depending on your overall income and other deductions. Depending on your income, your state tax rate, and your interest payment, one option might save more than the other.
