I don't think jurors are allowed to write anything during a case (one would think allowing jurors to take notes would be a good thing, but it's not without dangers). And I think both counsel have a right to see any notes passed to the judge.
Still, if someone were to write a note saying "Certain information has come to light during the trial that would make it impossible for me to evaluate all evidence impartially. What should I do about it?" I would guess that would be reasonable. Not sure how a judge would want to proceed then. I'm pretty certain both counsel would want to be present at any discussion between the judge and the juror, and it might be hard to explain one's causes for concern to the judge without giving information to counsel that could prejudice the case.
If a judge decides to drop a juror, there is absolutely nothing that either side can do about it. They don't get to know why the juror dropped, either. If the judge decides not to dismiss the juror, then there still won't be a contradictory hearing on the matter. The judge has already decided that the juror is staying.
As for being able to write or not write stuff, it depends on the court. Some judges allow it, some don't, some even provide pen and paper to the jurors.