In the United States, certain requirements to become a Notary Public vary by state. However, every state requires that a Notary Public maintain a written record of persons served. Most states also require that a Notary Public be bonded for a certain amount and commissions are not for life. Here in Arizona, persons applying to become a Notary Public must file all applications with the Superior Court in the county in which they reside. They must have a bond of at least $15,000 if I remember correctly and are subject to screening.
Notaries essentially serve the same purpose as court clerks in that they certify that the instrument is true and correct. Main difference is that court clerks can only certify instruments that are copies of originals on file with the court of record whereas the Notary Public only maintains the written log of documents Notarized and not original documents.
As for notarizing one's own birth certificate copy, it may or may not be legal and could certainly raise ethics concerns. Also, in most cases you would have to request a certified copy from the court or agency of record anyways, so notarizing the copy would only be a waste of time and money. As a deputy court clerk, we constantly get request for copies of marriage licenses and other similar documents such as divorce decrees or Clerk's Certificate of no record of previous marriage. Those cannot be notarized since the Notary cannot prove that they are True and Correct copies of the originals on file with the court. Also, those documents must bear the Seal of the Superior Court and the signature of the certifying court officer (in my case, a deputy clerk).
"When a man runs on railroads over half of his lifetime he is fit for nothing else-and at times he don't know that."