Yes you may and indicate USCIS error without fee.

I agree with my colleague - best to resubmit an updated affidavit of support.

The application rejects typing "/" to put N/A in non relevant gaps, shall I hand write it after printing or put "NA" instead

Yes you may indicate “n/a” where applicable.

I am in adjustment of status process, will getting COVID 19 pandemic unemployment assistance affect my green card ?

No this should not impact your pending application.

Does applying for Pandemic Unemployment Assistance (PUA) impact my citizenship process (naturalization) negatively?

No this should not negatively impact your case.

You will need to make every effort to return or your GC will be considered abandoned and a returning resident would need to be applied for.

Advance parole

Yes advance parole may be used to travel abroad while application is pending.

Yes USCIS may verify information about your bank account with bank.

Yes non-availability certificate, statements from witnesses who can attest to their knowledge of birth, or school or hospital records.

How can I get Authorization to continue to work?

You should discuss with your retained counsel re: next steps and NTA.