What exactly is an "adverb"?

The theory of adverbs (and of Conjunction Reduction) given by McCawley in The Syntactic Phenomena of English explains why you can't get your example

*"I'll go tomorrow and quietly."

It would have to come by Conjunction Reduction from

[[I'll go] tomorrow] and I'll [[go] quietly]

but Conjunction Reduction requires the two constituents to be conjoined to occupy the same place in the original conjoined structures. That is not the case here, as I've indicated with the brackets -- "tomorrow" is a sentence modifier, but the manner adverb "quietly" is a V' modifier.

Most anything that answers a “when” question can be roped into service as an adverb, even if it is normally considered a noun or a prepositional phrase.

Q: When are you going?

A: Immediately.
A: Soon.
A: Now.
A: In a while.
A: After I’m done eating.
A: Tomorrow.
A: Next week.
A: Friday.
A: Never.

All these answers are acting like adverbs in this context. But are they really adverbs? What about nouns like tomorrow or next Tuesday? Are those adverbs, too?

The simple answer is “Yes.”

A better answer is “Well sure, sorta.”

But the best is answer is “What’s an adverb?”

And thereon hangs a much longer tale.

The problem is, asking whether something is an adverb is a devilishly loaded question. It assumes that there is such a thing as a “real adverb”, which as it turns out isn’t a reasonable assumption at all. There are just words, and words do as they please — meaning, they do as their speakers please. _{^{(Yes, Humpty-Dumpty was right after all. :)}}

Sometimes they happen to do jobs we call adverbial, like answering “when” questions. So, for a short-hand, we call them adverbs there, using a classic part of speech tag known to scholars and school children alike.

Understand that part-of-speech (POS) tags are just an invention. They are sometimes a useful invention, true, but there are not really a necessary one. That’s because words in English are free to fall into whatever slot they want to, to do the needed job. That’s why we end up having so many “this as that” type tags when doing good POS assignment in natural language work on a computer.

One thing that occurs to me is that these “noun-adverbs” (meaning nouns doing an adverbial job) do not appear to admit normal adverbial inflections into the comparative and superlative degrees. Go back to the list of A: ... adverbial answers above and try to inflect them by degree. Sure, you can do something sooner, so that one inflects. But some do not. You are free to “do something tomorrow”, but you may not “do it *more tomorrow”.

Perhaps it bothers you that we have words doing one of the (many) duties of adverbs by answering temporal questions, but which refuse to be roped into another customary adverbial duty, inflections according to degree. Is that perhaps the origin of the question? If so, then the problem is really that we need more distinct parts of speech than the traditional ones.

One problem with assigning POS tags to English words is that this is something of an artificial distinction, the product of artifice alone. All that matters is how a word is used, and even then the granularity of your tag-set varies considerably. In short, it just depends how you slice it.

You will find that the POS tag-sets used by various reference works vary a bit, sometimes a good bit. Even the OED changed a little in how it assigns parts of speech to senses between v2 and v3. For example, many words once marked as a prefix or suffix in the OED2 are now held to be combining forms.

This is especially noticeable when doing syntactic analysis for natural language processing. The parser will make POS assignments to each word in the sentence analysed, and you have to know what each POS tag means.

A particularly common set of POS tags is the Penn Treebank tags. Someone who comes from the school that admits only the seven “classic” parts of speech (^{_{NOUN, PRONOUN, VERB, ADJECTIVE, ADVERB, PREPOSITION, CONJUNCTION}}) may find Penn’s 36 POS tags to be elaborate and useful. But I am not especially fond of them, because they conflate many things that are useful to distinguish in a parse. I prefer the NUPOS tagset, which is a much, much richer tag-set.

If you look at the NUPOS tags for adverbs, you will find that they have a category of adverb called a noun-adverb, meaning a noun used in a slot expecting an adverb, analogously to how a noun-adjective is a noun used in a slot expecting an adjective.

This isn’t anything fancy, and is indeed the very phenomenon we’re discussing here. When we say “Go home”, we find that we are using home, a word normally thought of as a noun, as an adverb. That’s because we are indicating where to go, and where is an adverbial application. If you like fancy words, locatives are always adverbs. ^{_{(And home is a very good example of a locative, and a very popular one historically just as it is today. That’s why the noun for home in Latin, domus, preserved a vestigial locative form, domī, but lost almost all the other distinct locative inflexions for the rest of its nouns.)}}

Where some classical grammars use 7 POS tags and Penn uses 36 of them, NUPOS uses 17 major word classes:

But those 17 are further split up into a set of 34, including things like this:

Even there, we can see that here there is such a thing as an adverb/noun, which belongs to the major class of noun. But there’s more than one way to skin a cat, which is why NUPOS goes much farther, dividing up those 34 major classes into 241 different final POS tags. Here for example are the adverbial NUPOS tags, with illustrative examples:

Yup, that’s a lot of POS tags. But it is useful for people doing NLP to have these nuanced distinctions. It may be useful in other work, too.

So which of those are adverbs? Hard question. Facile answer is that those beginning with av* are. Oh and dx. Maybe some others, too.

See the problem? We’re categorizing things according to their job in the actual phrase, and words in English are super-flexible in their job-duties, much more so than a dictionary’s simple-minded part-of-speech listing suggests.

Cool Elf is right: those are all adverbs, because they modify verbs, adjectives, or other adverbs. That is not to say that there aren't different kinds of adverbs; I believe modern Anglo-Saxon syntacticians even use different words for them. But this is the meaning of the word "adverb" as it is commonly used.

As to tomorrow, your example is a just a regular semantic syllepsis, which proves nothing:

I hit the ball and my head.

This doesn't mean that ball and head aren't both nouns. It just sounds odd because the verb hit is used in two slightly different ways.