No. In general - and with some exceptions - in English proper nouns are capitalized, while normal nouns are not. Proper nouns are names, like your name ("John", "Marie", or whatever); and names of specific individual items (such as "the Parthenon"), as opposed to generic names that apply to any of a group of things ("a person", "people", "a building").
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Yes
yes
No. It's not necessary.
No.
No.
No one can take better care than the parents on students.
You do, because they are proper nouns.
If a person is referring to the monarchy system of government, it isn't necessary to capitalise it. However, if the person is referring to the British Monarchy itself, then it is proper to capitalise it.
Greek Mythology or Greek Myth.
You can capitalise it if you want.
When you are referring to the planet Earth.At the beginning of a sentence.