Proof of citizenship and age, usually a birth certificate or passport will do. You will have to present them when you register with the state to be put on the ballot. You will have to contact your state to find out what the filing fees and required proofs are. In some places you have to have a petition showing that you have some support.
The general rule for signatures is that as long as you make some sort of mark that you can identify as your mark, then it is a valid "signature." Even an X is a valid signature as long as the person making the X can identify it as theirs. As long as the mark made by the person represents their intent to authenticate the document, it can theoretically be anything. In the movies of the old west one has seen where a person unable to read or write authenticates a document with simply an "X", the document is legal. That is the value of a witness who can testify that the document was in fact made by that person. The problem may arise later if the original signer or the witness become unavailable and there is no reliable sample for comparison. An "X" on a legal document such as a deed or mortgage would need to be witnessed on the face of the document in order for it to be a valid conveyance.
I don't know where you are and that can make a big difference in the answer. In the US, a notary is only witnessing the person signing a document and has no authority concerning the validity of the content of the document. In the US, it would be considered a legal division only if both parties of the dissolution sign it.