Yes, the reimbursement is taxable income. It's really just extra pay, no matter what the employer says it's for.
So I am paying tax on the premium twice since I have to pay taxes on soc sec income
No, you are not paying tax on it twice. You only pay the premium once. You either pay it with taxable Social Security income, or you pay it with taxable income from your job. You can do whatever you want with the other taxable income that you don't use to pay the Medicare premium.
If you have the Medicare premium deducted from your Social Security benefit, then you are paying it with your Social Security income. You are not using the extra money from your employer to pay the premium.
It depends. It may be that there is nothing to do.
Medicare part B and D premiums are carried to your medical expenses for itemized deduction. If you take the standard deduction, then you are not gaining a deduction so no need to include the medicare premiums paid. If you are itemizing your deductions and are able to take a medical expense deduction on Schedule A, then you would need to make sure the premiums are included when entering your medical expenses.
Some states allow a deduction for health insurance premiums paid as well. Check your state to see if you can use your medical expenses or health insurance.
[Edited: 04/01/2022 | 1:38p PST]