It depends on where you use it within the sentence.
No, because it is not a proper nou n.
"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.
Capitalize the "c" in the word congress when you're referring specifically to the United States Congress or the title of another specific congress.
You do not capitalize chickenpox in a sentence. The exception is if the word is the first in the sentence.
Congress is, member is not.
No, you do not capitalize "first birthday" in a sentence.
You only capitalize 'deceased' if it is at the start of a sentence.
The Library of Congress is Americas. The Ashurbanipal is Ancient Mesopotamia's. The Library of Congress is the Largest Library in the world. Ancient Mesopotamia isn't. The Library of Congress is more up to date then the Ashurbanipal Library. The Library of Congress has photos. Ashurbanipal Library doesn't. The Library of Congress was made for congress the Ashurbanipal Library wasn't.
No, you do not capitalize marine biologist in a sentence. (Unless marine is in the beginning of the sentence)
No.
Yes, you always capitalize a day of the week.