I think the term you are looking for is "recusal", that the judge would take himself off a case, but the facts of the case as you've stated them don't seem to support a motion for recusal. Recusal might happen, for instance, if a Judge represented a party in private practice before he became a judge or was related to a party. It's akin to "conflicts of interest".
From what you're saying, the same judge, presumably a Family Court judge, decided a neglect petition against you and is now sitting on a related custody petition. But, as far as I know, there's nothing wrong with that, and he isn't required to seek a new judge to replace him because he's already ruled against you or "biased" (in your eyes or a lay interpretation of bias, but not the legal meaning). He just doesn't like you, perhaps. Since there's no right in Family Court to have cases decided by a "jury of your peers", you are probably stuck with this judge as a factfinder and must rely on your attorney to explain why the facts of the neglect case don't bear on the custody matter.
I'm not sure I follow your point about being appointed and how that would bear on a judge's qualifications. Judges are usually elected in New York but can be appointed to fill out the unexpired term of another judge. Federal judges and the Supreme Court Justices are appointed with legislative approval. One system is not inherently better than the other or indicate any kind of "corruption". Good arguments can be made for both methods of appointment.
I agree with Jack. The judge can hear both cases. Unless you have a basis to have the judge recused, the you are stuck with that judge. Be careful if you ask a judge to recuse themselves and they do not , then you have an enemy on the bench.
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If the judte was appointed or elected, it doesn't make a difference. A family court judge has the authority to hear all cases brought in family court. Very often the judges will be assigned certain types of cases administratively but this does not take away their authority to hear other types of cases.
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