The preposition "on" has a number different meanings, one of which is to "physically be on top of something". The preposition in the sense of "on fire" is one of them (from dictionary.com):
13 . in a state or condition of; in the process of: on strike; The house is on fire!
One problem is that the entire concept of "part of speech" is very old. How we use it in English, especially in dictionaries, goes back to the study of Latin and Greek. In this view of English grammar "adverb" is the catch-all category where everything that doesn't fit into one of the other traditional categories ends up. (The others being noun, verb, adjective, pronoun, preposition, conjunction, and interjection.)
Now there is no one, true description of any language (except perhaps constructed languages such as yours). There are merely alternative or competing descriptions which appear over time as more independent analyses of the language are undertaken. Such descriptions or analyses may be called "grammars".
Most (but not all) grammars include a concept of word class under one name or another. So one problem is that "part of speech" has two meanings. One is the specific set of eight categories from the classical languages, the other is as a synonym for word class, which is a lot looser.
So all your example words are adverbs under this older stricter view of parts-of-speech, but their qualities and quirks can be much more thoroughly investigated in newer ways. And various new ways will have various new terms for the classes they put these various words into.
Unless you are inventing a new language specifically to embrace the classical parts of speech you don't have to worry in which they belong, but if you are inventing a new language to learn more about how language works then it will be worth your time reading up on the many newer grammars and language descriptions and analyses.
Best Answer
The phrase "there is" indicates the presence of the object of the sentence, without making the object the subject. The similar sentence "A book is on the table" would mean the same thing but restructures the statement to have a subject. Usually, when using the indefinite article "a", the statement will more often use "there is" to emphasize the presence of the book over the book itself.
The construct comes from the French term "il y a", literally translated as "that there has" but thought of by native French speakers as identical to "there is" (except conjugated using "avoir" instead of "etre"). As this shows, "there" basically replaces "il y" ("that there") from the French and so takes the place of the pronoun.
As for punctuation at the end of a sentence containing a quote, there are a lot of conflicting rules regarding punctuation in quotes. In American English, the rules are as follows: