Sometimes we believe a dying declaration because a person does not sin when he has nothing to gain (Rambam Mishna Archin 6:1). However, we do not believe it against a presumption (Baba Batra 134b and SA EH 157,7).
As for the three types of people who discovered that they are Jewish, the first must undergo giur l’chumra as it cannot be known if they married non-Jews during the last 500+ years (BTW, there are also many in Latin America, Spain and Portugal). Rav Aharon Soloveichik, however, wrote:
“They must be treated like full Jews in every way (counted for a
minyan, given aliyot, etc.).
Only when one of these anousim wishes to marry a Jew, must he or she
undergo full conversion. That is, he or she must undergo immersion in
a mikve (without the blessing) and full acceptance of mitzvot or
commitment to the Torah. A man, if he is uncircumcised, must in
addition undergo circumcision; if he is already circumcised, then he
has to undergo hatafat dam brit.”
There is an organization called The Association of Crypto-Jews of the Americas which works to re-integrate them in the Jewish community.
Regarding the latter, archives can be searched for documentation.