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Is it very rude to send in an application for a future term when the judge is not advertising that he/she is hiring for that term?

Most likely it will be disregarded. Small chance the Judge will be annoyed and will hold it against you, also a small chance the Judge will consider it and call you (larger chance if you have very impressive qualifications) but most likely it will be a waste of time and paper.

If you are, for example, 1st circuit or bust, is there a harm to just sending an app to all judges if you're a very competitive candidate willing to invest the time and money?

Unless the judge's posting says only apply via OSCAR, I think the chances of any harm are infinitesimal. Every now and then I've heard of a judge interviewing three candidates for one spot and being really impressed with two of them and making one of the applicants an offer for the initial term and one for the following term so there is a chance you're application will hit the spot with a judge and he or she will invite you to interview for a future term.