Could they make a weapon that could banish Jeanne, though? They made the magic arrow that trapped her soul, but most of the people who made that are probably dead by now, even saying they could make something that does the reverse. Maybe it 's like a bomb: once it's exploded, you can't very well unexplode it.
We haven't seen anything that the court can really do. While I have gotten the impression that the court is rather abandoned, I don't think that they have really lost their technical prowess at all (as you can see from the ethereal experiments they have been doing), although they may have lost their more direct magical knowledge.
The series has shown pretty clearly that besides Jones, everything can die. I suspect that even Coyote can be killed (although it would require god level weapons/enemies and him badly failing in one of his tricks). I don't see any reason Jeanne would be any different. While it might not be possible to reverse the process and banish her (although I think it would be possible (even if the court can no longer manage it)), killing her soul shouldn't provide much challenge to a appropriately armed Jones.
I don't think it would really be all that hard to make a weapon that could kill her either since she just appears to be a super-ghost (and that's discounting the fact that the court has probably accumulated some pretty powerful artifacts over the years). Actually being able to beat Jeanne would be tough without Jones (although I have very little doubt they could eventually manage it since Jeanne is unable to leave the canyon), but I have little doubt that Jones far outclasses her due to the whole "perfectly invulnerable" thing she has going on.
The reason why the court hasn't tried to fix it is because the founders and other people involved in it erased all information that she existed, or so they thought. Even though Diego managed to keep that shrine with the painting hidden and theres that camerabot, all public records of Jeanne were destroyed and the knowledge of the arrow thing died with the founders. Diego could have kept notes somewhere though.....
Also, Jones could very well be a body with a spirit, just not a human spirit.
All public record. I suspect that the headmaster knows exactly what Jeanne is and does, even if he doesn't know her origin. As you said, it doesn't seem that despite the efforts they made that all the information was erased.
I believe that the headmaster of the court knows about Jeanne (although he doesn't know about the kids attempt to kill her), and that Coyote gave them his tooth in an attempt to get them to kill Jeanne (but in such a roundabout way as to be able to deny that it was his plan to kill her (as well as the fact that she might be dangerous even to him)).
Note that there are a lot of suppositions of my own, quite a lot of what I said isn't actually backed up by any hard evidence.
E: We haven't seen anyone beside the teachers and members of the school staff (who indeed are rather lacking in knowledge as far as we can tell), but there rather obviously has to be others living there (eg. to run the ethereal experiments), and we have no clue what skills/knowledge they possess.