A hard copy of W-2c

I already e-filed my tax return and received the federal refund. However, NYS tax Department sent me a letter to check amount of tax that withheld from my income. When I checked my W-2 again, I realized that my SSN on W-2 is wrong. I e-filed my tax return with correct SSN though. I had to respond to NYS Tax Department ASAP, I just did without mentioning wrong SSN on W-2. I’m still waiting for status update on state tax return. 

Now I have some questions:

- My former employer who issued W-2 with wrong SSN will produce W-2c soon, and told me that they will provide a PDF version of W-2c online but won’t send me a hard copy of W-2c in mail. Is a PDF version of W-2c enough for me as my record?

- If NYS Tax Department noticed my SSN is wrong on W-2 and rejected my tax return now, do I have to refile with W-2c still by April 15th?

- If my NYS tax return was successfully done without having problem with SSN ( I heard that often time tax return can be done even if SSN is wrong on W-2 and correct SSN on application), what do I need to do after my W-2c is produced?

 

I’m living outside of the US and also in this severe Coronavirus situation, I’m worrying if I can get my state tax return...

Thank you very much in advance!!