The signature of the insured is only allowed to be signed by the insured under small circumstances.

Is it necessary for the insurer or the insured to fill in the contents of the insurance contract in person?

What needs to be filled in an insurance contract needs to be filled in by the insurer or the applicant himself, especially where the applicant signs.

Both the insurer and the insured are locals. Do I need to sign a supplementary insurance contract?

The insurer (you mean the insured) is a formal text contract, and the supplementary terms should also be in the text contract!

Should the applicant of supplementary insurance contract hold it?

The insured holds it, or the underage insured holds it.

You can send me MM directly if you have any questions.