Is 'there' an adverb or a preposition? (Or something else entirely!?)

First a caveat: you cannot always reliably test what function a certain word or phrase has by replacing it with some other word or phrase. However, normal adverbs serve the same function as many prepositional phrases do, so what you said about how it can replace in Paris supports treating there as an adverb.

A preposition occurs before a nominal phrase, like a noun; but there Paris is not possible, unlike in Paris. And so it is not a preposition, unlike the preposition in.

There is often called an adverbial demonstrative pronoun. While it does normally have an adverbial function (it describes where something happens), it has an antecedent: it refers back to a place that was mentioned earlier or that the listener or reader knows is relevant. So in your example, it refers back to Paris, a place; in its capacity of referring back to an antecedent, it is pronominal, functioning like a pronoun.

It is demonstrative because you can point at something while saying it, just like the demonstrative (personal and adjectival) pronouns this, that, those and these: I see that [pointing at object], we went there [pointing at place] etc.

Note that, in certain idiomatic constructions, there has almost evolved into something more like a particle, as in there was noöne in the room.

The words straight and right in your example are a bit complicated; they could be analysed in different ways. The two most obvious ways would be to either classify them as adverbs modifying there, or as adverbs modifying the verb in parallel with there. The simplest structure of I flew straight there would be the latter: how did you fly? I flew straight, not via some other point. Where did you fly? There. Those are two adverbs, one of manner and one of place/destination.