All of your sentences are valid and usual French.
The third sentence indeed negates the first one :
(1) Il n'y a qu'un livre: there is just one book (i.e. there are no more books than that one)
(3) Il n'y a pas qu'un livre: there is not just one book (i.e. there are more books than that one)
In the second sentence, plus isn't there no negate anything but to ...
The ne is optional is spoken French. You might as well drop le or both:
Ma situation financière est pire que je ne pensais.
Ma situation financière est pire que je pensais.
The feminine is not used, at least in modern French. The pronoun le refers to the whole phrase because the sentence does not say penser la situation pire but penser que la situation ...
No, ne does not create a double negative in French, no matter what is paired with it.
Rather, the pair ne ... pas is only one negative. You can substitute several other items for the pas, and you'll get a different "flavour" of negation with a different meaning.
This answer contains a partial list of negatives, including ne ... rien.
Note that ne ...
You can hear rien peut arrêter ça in spoken French with the expected meaning.
In written/formal French, a negation is built with a split negative form composed with the adverb ne and a pronoun like rien, pas, personne and the likes.
These pronouns come from nouns with a (possibly lost) positive value (rien=something, pas=step, personne=person) so this isn't ...
It's a formal and somewhat literary construction. First of all, il est (impersonal) for il y a (there is) is a formal construction. Then you have this literary expression "il n'y a pas jusqu'à/il n'est pas jusqu'à + substantif et proposition relative au subjonctif avec ne", "marquant avec insistance le point de référence limite" (TLFi) i....