First of all you should call your lawyer and ask him what your legal options are. In the USA Federal law requires courts to make reasonable accommodation for ones religious beliefs, and delaying a phone call a day or two should be considered "reasonable accommodation". Also, you should contact Agudath Israel or the OU, as these and similar organizations have extensive experience dealing with situations where demands of the courts conflict with religion. However it may be difficult to reach them on such short notice on Sunday erev Yom Tov.

If all else fails, you may ask a non-Jew to place the call on your behalf on Yom Tov, hopefully this will satisfy the Court's demands.

According to Rabbi Bess of LA's list Nyquil Children's Cold/Cough and Nyquil Liquid Multi-Symptom Cold & Flu Relief (Original, Cherry) are acceptable for use on Pesach.

First of all Reb Moshe never permitted a davar hamaamid added by a goy, because davar hamaamid is not batel. What he did allow was allowing a goy in his factory to add an insignificant amount of non-Kosher ingredient which is batel l'Halacha, where the only issue is ein mevatlin issur l'chatchila which does not apply to a goy. Even regarding this Reb Moshe himself wrote that while it is permitted according to Halacha, it is not proper to do (mechuar hadavar).

To the best of my knowledge all reputable hechsherim will only certify a product where all its ingredients are Kosher and will not rely on this kula of Reb Moshe by itself; certainly this is the OU's position. However under certain circumstances the will use Reb Moshe's heter as a mitigating circumstance, such as if the method used to kasher the machinery is questionable and subject to a machlokes, if the factory is non-Jewish owned and at least 24 hours have elapsed since the equipment was last used for non-Kosher products they may at times rely on this questionable process where they would not allow a Jew to rely on it.

According to the dictionary Tapioca is a starch extracted from the root of the cassava plant. Therefore it should be permitted even for Ashkenazim and not be considered kitniyos. However it would certainly require a reliable Pesach hashgacha.