No.
"Nope." Improved answer: Normally, no. The library should be written as I have just done, without capitalization. However, if you are talking about a specific library and using its title, such as "The Central Public Library of Nowheresville," you should capitalize the word library within the title. On rare occasions, it is okay to capitalize "The Library" if you are talking about "the" library in the U.S. - the Library of Congress. But even that is not generally accepted.
No, you should not. You should only capitalize a word when it is at the start of a sentence or when it is a proper noun.
Yes, you should capitalize the word after a comma in a sentence.
Yes, you should capitalize the word "Corporate" when it is part of a proper noun or title. For example: Corporate Headquarters.
Yes
Yes, you should capitalize the word "scurvy" if it is the first word in a sentence or if it is part of a proper noun or title.
Yes, you should capitalize the word "Figure" when referring to a figure in a paper.
Yes, if you're referring to the concept of Fate as a proper noun, then you should capitalize it. For example, in the phrase "believer in Fate," the word should not be capitalized.
In a title, you should capitalize all words except for short conjunctions like "for," "and," "but," and "or" unless they are the first or last word in the title.
You only capitalize with if it begins a sentence. Example:With recent rains, crops should grow well.
Unless it is the first word in a sentence, then you don't need to capitalize "tobacco."
No.