england-and-wales
In theory, could the creditor say/post anything they like about the
debtor with no repercussions before agreeing to the settlement?
In theory, yes (but see Dale M's answer). You'd want to retain good evidence which shows the timing of the disclosure so you can prove it took place before the agreement.
Note that if you disclose information and fail to tell the other party about it, you could be at risk of misrepresentation. The other party might argue that in entering into settlement negotiations over an agreement that contains an NDA, you made an implied representation that you haven't already disclosed the information.
If the settlement agreement contained a confidentiality term that the
agreement itself is confidential, could the creditor share copies of
it with other people?
Usually NDAs contain clauses which provide for exceptions e.g. for disclosure to legal advisers, courts, law enforcement officers, etc. Failing that, you shouldn't share copies with other people or you risk being in breach of contract.
Would those other people be bound to confidentiality in any sort of
way?
A third party cannot be bound by a contract that they were not a party to. Depending on the nature of the third party and the information, there might be other legal mechanisms which prevent disclosure. For example, a lawyer would be prevented from disclosure by their regulator's code of conduct. In the majority of other cases, the third party would be free to disclose the information.
For this reason, clauses which provide for disclosure exceptions usually also require that if you use the exception, you agree to sign an equivalent NDA with the third party and to guarantee that they won't disclose. In this way if the third party discloses, you can still be sued for breach of contract.