It may or may not be unairworthy (some paperwork is required by the type certificate, other paperwork isn't), but if it's paperwork required by regulation, the operation would be illegal, just under a different section of the FAR's.
Probably not, unless the FAA can show that the bad paperwork directly affected the safety of that flight, and the pilot either knew or should have known that.
If the pilot had the paperwork corrected per the regs once s/he discovered the discrepancy but before further flight, I don't see how there would be a problem. However, if the pilot altered documents which pilots are not authorized to sign (like maintenance records other than preventive maintenance -- E-AB issues aside), I see the potential for enforcement action for violation of any or all of 91.7, 91.9, 91.103, 91.203, 91.207, 91.411, 91.413, or probably several others depending on just what the discrepancy was. Note that FAA Order 2150.3B ramps up the sanctions and eliminates administrative options like remedial training if pilot knew s/he was in violation and flew it anyway. Further, altering documents signed by someone else could also result in 61.59 violation, and that almost invariably results in revocation of all certificates.
Now, if we're back to the business of pilots altering their aircraft's weight and balance records to correct an error by the authorized person who prepared and signed them, I think the FAA would talk with the pilot to see if s/he understood the rules on who's authorized to prepare and sign aircraft maintenance records including W&B documents such as the empty weight/cg data and the equipment list. If the pilot didn't realize that requires an A&P's signature, I think the pilot would be counseled and told to get it properly signed by an authorized person before further flight. If the pilot refused to do that, I think enforcement action might ensue.