No Stupid Questions

"Corrupt" would almost certainly be a statement of opinion, so not actionable in the US. A lot more detail would be necessary for this to be defamation.

"Judge XXXX has taken millions in shadow bribes and has consistently ruled for the wishes of his/her benefactors. There has been a history of being reversed on appeal proving their bias. Also I watched them kick a puppy."

Then, obviously, these things would have to be false. Even then, the bar is pretty high. There are exceptions both ways on this, but as a general guideline, if the public knows a person's name (judge in a high profile case, for example) they are probably classified as a public figure. The rule there is one of "actual malice" which isn't exactly what it sounds like, but it's the highest bar for defamation cases.

The speaker would have to say something factually false, knowingly or with no regard for the truth. Giuliani, for one recent example, was found guilty of defaming the Georgia election workers, because he went into great detail about his false claims, and he was told repeatedly that thise claims were false, but he kept going.

outside court

If you mean a different time, not while the proceedings are ongoing, then it's easy: yes, the judge has the same right to sue as anybody else. (And No, that would not make the judge biased. That idea is somewhat ridiculous).

If you mean during the proceedings, then the judge has the additional right to call that person to order with a fine. Repeatedly, if needed.

No matter what the outcome, it's probably a poor choice to do this before your sentencing, which the judge you're slandering has full control over.

IANAL. Probably not.

Defamation needs to show damages and the judge would need to show that somehow those statements materially damaged them.

Public figures also need to meet higher bars for defamation.

The short answer is "yes" if you define "suing" as filing a lawsuit with a court. Anyone can file a lawsuit in court for pretty much any reason. Whether or not it goes anywhere depends on the strength of the claim and other factors.

The longer answer is still "yes" if you narrow your definition of "suing" to filing a legitimate lawsuit with the potential to be successful. Judges are still people and can suffer real damage due to libelous or slanderous claims. There are certain elements that must be proven to be successful though. One of the defenses is for rhetorical hyperbole. As a hypothetical example, if a defendant with a penchant for grandiose and bombastic hyperbole and known for calling everything and everyone under the sun "corrupt" because he's a pathetic baby calls the judge corrupt, it would likely fail based on the comment being rhetorical hyperbole. It would be similar to saying someone "killed" another person when no actual murder took place and it was like a verbal smack down.

If a judge is called 'corrupt' by a defendant

After thinking again:

In most parts of the world, such behaviour would bring you a few years extra in some really bad jail, or 30 good strikes extra with the whip (this is near death), or simply a stronger kind of penalty than you deserve for whatever you have done. It is a bad idea. It is far from funny.

Of course you are talking of the Usa here. And of course their system is corrupted. But I really don't think it is that corrupted that you could actually get an advantage from such bad behaviour instead of an extra penalty.

Asking for no particular reason?

As with everything it depends. The judge can file but unless they can show they suffered loss because of the comment another judge would throw the case out due to lack of standing.

They can always sue for defamation I guess, depending on the laws of whichever country

I’m guessing that could give the defendant ground to appeal the original ruling because the judge was biased by the alleged libel.