This is a claim of right. See here.
https://ttlc.intuit.com/community/unemployment-taxes/help/what-is-a-claim-of-right-repayment/00/2153...
1. No. Use the claim of right procedure for whenever you actually make the repayment (sounds like you made the repayment in 2022 so you will claim the credit on your 2022 tax return).
2. No. Use the claim of right procedure on your 2022 tax return. For FICA and Medicare, you need to file a separate form 843. For Georgia, you will have to look up their procedure. (see below)
3. No. Your 2021 W-2 shows what you were actually paid during the 2021 calendar year and can't be changed by something that happens in 2022.
4. No.
"I can ask I ask the employer to eat the taxes paid or any losses I incur during this process (filings costs, CPA, taxes etc)?"
You can ask anything. Whether you have a right to compensation depends on your contract and state employment law.
"What's the easiest and cheapest way to handle this?"
You will need to use Turbotax installed on your own computer for 2022 (PC or Mac, from a CD or download) and not the online version. Any version of the desktop program that is appropriate for your other tax situations will allow you to make a claim of right.
"Also, does the same principle also apply to joining bonuses paid which need to be repaid back or is that different because this an employers error and the joining bonus repayment is a contractual agreement."
There are two ways to claim a wage repayment over $3000, as a special itemized deduction on schedule A, or as a tax credit that represents the amount of tax you would not have paid if the money had not been included on your prior tax return. Generally, if your income or tax rate is higher now, take the deduction, if your income or tax rate is lower now than when you paid the tax, take the credit.
Anyone can take the deduction for a wage repayment. However, you can only take the claim of right tax credit if you had a reasonable belief at the time you received the money that you had an unrestricted right to the funds. This obviously depends on each person's circumstances. For example, if you got a bonus that was contingent on working 3 years, and you knew you were leaving after 2 years, then you knew that you did not have an unrestricted right to the money. Or, if someone gets overpaid and they know it, and they keep quiet hoping the employer won't notice, then they knew they did not have an unrestricted right to the money.
For Georgia, they probably have a claim of right procedure, but I don't know it since there are so many states with different rules. You have a year to figure it out since you will claim this on your 2022 tax return.
For social security and medicare tax, you need to file a separate form 843. You will need proof of the overpayment/repayment, and you will need a letter from your former employer where they state that they will not be refunding the excess social security and medicare. You can file form 843 at any time, it is not part of your regular tax return and is not included in Turbotax.
https://www.irs.gov/forms-pubs/about-form-843
