Your allowance is supposed to run through Payroll and is taxable through payroll, unless it meets one of the IRS criteria. It also is supposed to be reported on the W4, in one of the boxes, such as 12 or 14, with the appropriate IRS code. Your employer is doing this incorrectly.

What you do with that money is up to you. Basically, your employer is giving you a Bonus = allowance, and their justification for giving it and the amount, is based on the discussion you had with them regarding health care benefits.

HSA is something you Qualify for, because your health insurance coverage is a High Deductible Health Plan. It has no relation to how you get any funds you intend to use for health.

There is no such thing as "1099- Misc expense" unless you are Running a business, and the reason a customer sends a 1099 relates to that income reported to you. The expenses associated with that business income would then be part of your tax reporting, typically on Sched C. As an employee, you are not Also supposed to get a 1099-Misc.

Oh, for this: "Similar to mileage and other self-employment expenses."

You don't have either of these, if you are paid through payroll and get a W2 from your employer. W2 = you are not self-employed. You are the employee of that business.