Assuming it's in federal court? Probably a 12(b)(6) motion to dismiss for failure to state a claim — essentially the defendant would say "your honor, even if everything my opponent claims is true, there's no valid legal claim here." Not a lot of Latin terms involved. (If it's in California state court — because, f/x, they're both residents of California — it's called a "demurrer," which is antiquated, though not Latin. I think a few other states still have demurrers.) It might also get tossed out under Rule 12(b)(1) for lack of subject-matter jurisdiction. Whatever attorney they suckered into bringing it could also be subject to sanctions, which would go up significantly if you followed it with a frivolous appeal.
What if whoever filed it tried to argue that losing the Atlasia election caused them emotional distress or something and thus sued for damages?
You can't get into court just by claiming something caused emotional distress, you have to show that the facts of the incident you're suing over "plausibly gives rise to an entitlement to relief." It is fairly implausible to say that an online dispute over a fake election involves the kind of "outrageous and intolerable conduct," for instance, that's necessary to succeed on a claim for intentional inflection of emotional distress.
Also did any attorneys get sanctioned in the past for filing say, Birther lawsuits or those joke lawsuits against God?
I believe Orly Taitz got sanctioned in connection to the birther stuff.