Partially. If he was considered Jewish and then says that he “converted himself” he may not marry a Jew as he made himself prohibited
(???? ????? ????? ???????) but his children may ( SA YD 268:11). However, if the bet din knows that he went to the mikva for tevillat Ezra or she went to purify herself from nidda they are considered Jewish b’diavad (ibid seif 3). If he claims that abet din converted him he is believed because of a ???? (ibid Shach seif katan 20).
Regarding a case where a doubt arises see ??? ????? ????? ?????? ?????? on the website of Yeshivat Shaalvim. However, in general we have a rule that ?? ??????? ????? ?????, ????? ???? ??? ??????? (SA EH 2:2) so if there is no basis for doubt there is no need to check and perhaps it would even be heretical to do so (Pitchei Teshuva YD 116:10).