What is the ruling on a mother saying, "The Prophet has watered you [meaning, may the Prophet quench your thirst]," and is there an issue with it? And does the word "God" take the ruling of the Most Beautiful Names of Allah, and is it permissible to burn it for fear of desecration?

The statement, "I implore you by the Prophet," is impermissible because it constitutes seeking intercession from the Prophet (peace be upon him) after his death, which is an innovation (bid'ah) not found in Islamic law. It also holds no reality, as the Prophet (peace be upon him) will not respond to such a plea, and the statement is meaningless.

It is obligatory to respect the names of Allah Almighty, regardless of the language in which they are written. You have done well by cutting out the word that means "Allah" out of fear of its desecration. There is no harm in burning it or burying it in a pure place.

If papers contain Quranic verses, prophetic traditions (hadith), or names of Allah, they must be preserved, or burned, or buried in a pure place. If a foreign word does not intend to refer to Allah unless it begins with a capital letter, then this characteristic is what dictates the ruling of preservation and respect.