for what? anyway, not usually.
Not necessarily. Just because it's notarized doesn't mean it's a public document or filed anywhere.
A postulate or axiom is an accepted statement of fact.
What kind of statement is accepted without further justification?
A logical argument in which each statement is backed up by a statement that is accepted as true is a proof.
Custody is determined by the courts.
It may be possible in some states to obtain a tattoo even if you are a minor provided that you have the permission of your parents to get the tattoo. Most commonly, a parent would need to be present. In certain states, it might be sufficient for a minor to turn up to a tattoo shop with a notarized statement. In that case, the notarized statement would be a written statement by the minor's parents giving the minor permission to be tattooed. The purpose of notarization is to ensure that the note is not forged, e.g. by the minor himself.
A statement that is factually correct and does not mislead.
Yes, I am a notary and courts accept notarized documents.
It can be. It is up to the prosecutor or the defense lawyer. They need to be notarized.
In federal court, the answer is no. I'm not aware of a notarization requirement for state court complaints, but state court rules could require it. In Pennsylvania state courts, a complaint needs to be "verified" but this does not require that it be notarized.