I don't think it's required. Largely the nat certificate is used to prove your US citizenship, which can now be done with your passport if you have one. It's best to order another nat certificate if you lost it just to have for use in the future, or in case USCIS wants to see it for some reason, but you can use your passport to prove your citizenship. That said, there are a couple questions on the immigration forms which ask for nat certificate-specific information (the number, place of issuance) which you'll need to answer, so if you have that jotted down somewhere, it shouldn't be too much of an issue. Otherwise, probably best to order before going further. The process usually takes 4-7 months, but it can vary greatly.
The answer given here is based solely on the facts provided, which may be incomplete, and is intended for general information only. It does not establish an attorney-client relationship, nor should it be relied upon for further legal action, which should only be taken after first having a consultation with an experienced, licensed attorney.
A valid US passport issued in your name is MORE than sufficient proof of your US citizenship. You do not need to provide the copy of your certificate of naturalization. It is usually either, or, but not both.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
No, it is not mandatory. A US passport is also proof of US citizenship.
Mr. Shusterman's (former INS Trial Attorney, 1976-82) response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.