Difference between an adverb modifying an NP consisting a single noun, and an adverb modifying a noun

Solution 1:

On the face of it, your sentences 1 and 2 seem extensionally identical -- that is, each is true if and only if the other is true. So I don't know whether I can find an answer. But I'll discuss it.

Syntacticians think about sentence structure in a peculiar way -- a way that traditional grammarians usually do not think about it. Syntacticians take sentence structure as something that actually exists and can be discovered, while other people generally think of it as a mere taxonomy -- a convenient classification system for discussing grammar. So it is often difficult for me to explain to a person trained in traditional grammar what the difference between a noun and a noun phrase is. (Hereafter N and NP.)

To a traditional grammarian, it's straightforward. A NP has to have more than one word in it, because that's what "phrase" means. A syntactician who has experience writing phrase structure grammar doesn't look at it that way at all. The difference between NP and N has nothing to do with the original source of the "P" in the name "NP". If I want to ensure that an NP always has more than one word, I have to make sure that I always write rules like NP -> old men but that I never write a rule like NP -> men.

This is not easy, when I want to start generalizing the rules. If I write NP -> Det N, I have to make sure other rules expand both Det and N as at least one word each. If I want to allow for sentences that have one word subjects, I can't have just the rule S -> NP VP -- I have to add the rule S ->N VP.

After a while, you start to wonder what the point is. Do one word subjects display any grammatical behavior distinct from that of multi-word subjects, other than having just a single word? Well, no.

There is also a difficulty with two tempting generalizations about English coordination. Ordinarily, you can only coordinate things of the same grammatical category, but then what about "socks and old shoes"? If a single word is of a different category from a multi-word phrase, this example should not be allowed. Furthermore, ordinarily the coordination of two like categories is of the same category as each of the things coordinated. The coordination of two verb phrases is a verb phrase, e.g. But an exception would have to be "socks and shoes", if I distinguish between one-word and multi-word categories, because I coordinated two single words and got a multi-word category.

So that's a problem with the sentence 2 in your question. You've proposed a difference in grammatical behavior that depends on whether a constituent has more than one word in it. But so far as I know, English just doesn't work this way, though it's not impossible that it might, I suppose.

Solution 2:

• The work is mostly Kim's.
• Only Kim resigned.

Aren't these examples of adverbs modifying nouns (which they are not supposed to do)?
Isn't mostly modifying Kim's, and isn't only modifying Kim?

In order,

1. No, these aren't examples of adverbs modifying nouns.
   Because only and mostly are not adverbs; they are Quantifiers.
   Quantifers are one kind of Determiner (other determiners include articles and demonstratives).
   Determiners are a part of noun phrases and generally appear before adjectives in English NPs.
   They can form quite complex phrases, just like any other part.

   • [the]D [big old brick]A [house]N
   • [several of the]D [big old brick]A [houses]N
   • [only a few of the]D [big old brick]A [houses]N

2. If the "not supposed to do" part of the question comes from an English grammar book that tells you what adverbs are and are not allowed to do, throw away the grammar book.
   Clearly the author was innocent of English grammar.

3. No, modification is more complicated than just being in front; only does not modify Kim.
   Only is a quantifier over the whole clause Kim resigned, and it has a focus on Kim.
   That's different from modifying Kim.

Quantifiers are often mistaken for adverbs because they're often used as adverbs in English.
A process called Quantifier-Floating (Q-Float) can move a quantifier to a number of positions
in an English sentence, so they can appear in many of the same positions as adverbs.

• The cat only puked once, in the living room, on the carpet.
• The cat puked only once, in the living room, on the carpet.
• The cat puked once, only in the living room, on the carpet.
• The cat puked once, in the living room, only on the carpet.

Note that these do not all mean the same thing. Some are ambiguous, some aren't.
This is par for the course with quantifiers and negatives (only is a negative trigger).
And there would be more variants if stress were considered, like the difference between

• The cat only puked once, in the living room, on the carpet.
• The cat only puked once, in the living room, on the carpet.

which disambiguates the focus of only.