Learn English – Extremely silent

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.

What you are doing in the examples very unique, more perfect is using an adverb to intensify a strong adjective; it seems a fairly good rule of thumb that this is often stylistically inadvisable. There is a certain inflation of the intensity of the adjectives, and the adverbs can be accused of pleonasm (as in "a triangular triangle"). This rule is often referred to on both sides of the Atlantic, and in many other languages too.

However, your other examples do more or less the opposite: you use adverbs to weaken (or qualify) the adjective in almost exactly, virtually all, and practically infinite. That is quite different: there is no inflation of the intensity of the adjectives themselves, nor are the adverbs pleonastic.

Other concerns may arise, though, in the case of absolutes and classifications (I like Fumblefingers' categories). Sometimes it is not very satisfying to qualify the unqualifiable, as in she's a bit pregnant: she's either pregnant or she isn't, so it is probably better to say she looks a bit pregnant, if that's what you mean. I think that's almost exactly the same thing, while perhaps acceptable, is usually not to be preferred over that's almost the same thing: the adjective feels a bit wordy and is made redundant by the adverb. The examples with all and infinite are what Fumblefingers calls absolutes, which I think should be qualified without qualms, where appropriate.