Yes
No, "hospital system" in general is not capitalized. However, if you are referring to a specific hospital system by name (e.g., "Mayo Clinic Health System"), you would capitalize it.
Yes, "World's" should be capitalized in the phrase "The World's first hospital" as it is a proper noun referring to a specific hospital.
It is not necessary to capitalize "hospital administration" in this context unless it is the official title of the degree or program. Instead, you can simply say that the person earned a master's degree in hospital administration.
The word "patients" should be capitalized when it is used at the beginning of a sentence or when it is part of a title or heading. Otherwise, it should generally be written in lowercase.
In "Yours truly," you capitalize the first letter of each word if you are using it as a complimentary close in a letter, as it is a formal sign-off.
no, but you do capitalize mum
You capitalize the first A but not the last a.
You capitalize proper nouns.
No, you do not capitalize the word drama.
You capitalize Huntington
do you capitalize the word protestant