Assuming that you have never lost your permanent resident status, you don't need to have a new green card in order to apply for naturalization. But considering your problems with getting a replacement green card, it would be a good idea for you to request a copy of your immigration file from the USCIS and see what's in there before filing for naturalization.
You might need to file a freedom of information act request to get your immigration file to see what has been going on with your case.
Legal disclaimer: This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Do not rely on this advice without speaking to an immigration attorney in detail about your case. This message does not create an attorney-client relationship.
Supposing you had a 10 year card (permanent, not conditional), you don't need to renew your card to apply for naturalization. I agree that it would be important to request a copy of your file before thinking about applying for naturalization.
This is general advice, and does not constitute an attorney client relationship.
Yes, I agree. Background checks were very seriously delayed around that time period. In about 2002-2003 some people got stuck literally for a couple years. Since you seem to know that your fingerprints will match the ones one your record, then I would certainly start with a request under the Freedom of Information Act for your file. The other option would be to apply to naturalize, and if something comes up address it then. FOIA will take longer, applying to natz without knowing what's in your file is riskier in case it turns out there was some other issue.
This reply is intended only as general information and does not constitute legal advice in any particular case. This reply does not create an attorney/client relationship.