You need to sit down with somebody who can work through the issues with you.
Based on your earlier question: Social Security and Medicare taxes not withheld from 4 years ago you have known about this issue.
The W-2 is your receipt from your employer, it reflects what you see when you total all the paychecks you received during the year. In fact the stubs for each check give you an update year-to-date accounting. There was no reason to wait until you get a new W-2 to talk to HR/payroll.
Because the previous employer didn't withhold SS and medicare you will have no credited income towards qualifying for Social security and medicare for those years. That will impact your monthly income when you retire.
Whose responsibility is it for paying SS and Medicare ? I read in some
forms that it is the company and not the individual.
The company must withhold the tax. The company must send your portion and their portion to the proper tax office. If they withhold but don't send the money they have major tax and legal issues.
But what if they thought based on your actions that you were exempt? That is a legal issue. What role did you play, and what is your exposure? You will have to talk to a tax lawyer.
The IRS till date hasn't questioned me or the previous company.
I have no idea how the interaction between the IRS and the Social Security Administration work regarding W-2s with no SS and medicare withheld. How does the IRS know that is correct?
I don't know what you will owe, and if you do how far back they will go.