In a formal writing, yes. In an informal writing, you don't have to.
The full name, "Anorexia Nervosa", should be capitalized. The simple and more common usage of "anorexia" does not need to be capitalized.
Yes, you should capitalize the word "Figure" when referring to a figure in a paper.
No
Yes, always capitalize a state.
If you are writing in APA style (and I assume you are), you do not capitalize the names of theories or hypotheses.
No they shouldn't be capitalized.
You should capitalize island when it is a part of a proper name. For example: Long Island, Gillian's Island.
To format a journal title in a paper, you should italicize the title and capitalize the first letter of each major word.
It should be--- The first paper asked the question, " Why attend such a conference?" with the answer, "Connections."
The scientific name for anorexia is Anorexia Nervosa.
It should be--- The first paper asked the question, "Why attend such a conference?" with the answer, "Connections."
It should be--- The first paper asked the question, " Why attend such a conference?" with the answer, "Connections."