The term "Esperanto" doesn't follow the standard rules solely because it wasn't intended to be used this way. Zamenhof (the creator of Esperanto) originally titled the language "Internacia Lingvo", international language. However, he authored the book proposing the language under the name "Dr. Esperanto", and this people adapted into the name of the language. I believe "esperanto" actually means something like "seeker"
I believe it's due to the fact Esperanto is a nickname, derived from LLZ's pseudonymous handle, rather than the actual name of the language.
Perhaps if someone founded a country called "Esperantio" then the language could become "la Esperanta"?
As for the proper name, I dunno. I've seen very few people refer to it as "la lingvo internacia". Perhaps someone better versed in Esperanto and Esperanto history could provide something more substantive than me writing off the seat of my pants.
Adverbs in Esperanto often correspond to prepositional phrases in English. So Esperante could mean "in Esperanto."
- Mi parolas Esperanto - I am speaking in Esperanto.
- Mi parolas Esperanton - I speak Esperanto.
An adverb here answers the question "how do you speak?". An object noun (ending in -on) answers the question "what do you speak?".