I know that we can say the following sentences.

1. The school was very different.
2. The school was much different.
3. The school was a lot different.

Then, can we say,
4. She was much happy.?

It doesn't sound right, but I wonder why.
Both "different" and "happy" are adjectives.

Apple

"Very" is an all-purpose intensifier that can be used with adjectives that are gradable. ("Happy" is a gradable adjective; "eternal" is not. It's non-gradable, absolute; something is either eternal or it's not.)

"Much" and "a lot" are used in comparative utterances to mean "by a great amount." "Much" and "a lot" can be used with "happier" or "less happy," but not with "happy" alone.

"Different" can be modified by "very," "much" or "a lot." "Different" is a kind of comparative, and as such it requires another entity with which to be compared. The rest of the comparative phrase is either expressed or understood from the previous context. It is, in other words, identified. Whether or not the other member of the comparative phrase is expressed in the utterance, it's always implied. Look at this example:

"” When we visited the high school's athletic facility we found it the same as it had been twenty years ago when we were students there. The school itself was very/much/a lot different [from how it had been twenty years ago]. The classrooms we had known had been converted into various lounges and cybercafes, and all academic instruction was done in new computer centers with robot instructors called "instrubots...."

We can say the same thing about "similar." There's always something or someone else out front or lurking in the discourse to which the noun is, or is not, "similar."

Marilyn

Very interesting. Thank you. Let me confirm if I got the idea right.
There are some adjectives that are not gradable. They are absolute, such as eternal.
My question: Is "alone" an absolute adjective like "eternal"? How about "the same"?
Are there any others?

Gradable adjectives cannot be intensified by "much" or "a lot" unless they are in comparative forms. So "very happy" is acceptable but, "much happy" and " a lot happy" are not.

"Different" is itself comparative, because it implies there is something, or someone else to be "different". Similar is another example.

Is there a way to see this in dictionaries? Is it stated in grammar books?

Apple

Let me clarify. It was an oversimplification to imply that there are only two kinds of adjectives "” gradable and absolute. Actually, there are no airtight categories that can't be breached. (The classic example of a nongradable, absolute adjective is "pregnant," but even this one is occasionally made gradable in an utterance like "She was very pregnant when she went up onto the roof to get the cat.") "Alone," under most circumstances, is absolute, but one could say

"” Herbert felt very alone without his pet chicken Flossie ("Much" can't occur alone here, but it could be part of the adverbial phrase "very much.")

"The same" is different; it's a comparative. You can say

"” He was furious and I felt [very] much the same ("Very" modifies "much.")

Here's a brief look at what the Cambridge Grammar of the English Language* has to say on the subject:

"The degree [of a gradable adjective] can be questioned or indicated by means of a degree adverb: [How good] is it? She seems [very young].... There are, however, a great many adjectives that are non-gradable. The following small sample will give an idea of how extensive the class of non-gradable adjectives is:

latter, equine, medical, federal, glandular, tenth, utter, residual, marine (abridged list)

"It should be emphasised, however, that the distinction between gradable and non-gradable "” like that between count and non-count in nouns "” applies to uses or senses of adjectives [italics mine] rather than to adjectives as lexemes [words found as entries in dictionaries]. Compare:

"the public highway/a very public quarrel
a British passport/He sounds very British
The door was open/You haven't been very open with us [...]

"On so-called 'absolute' adjectives

"Adjectives such as the following are traditionally classified as 'absolutes':

absolute, complete, correct, equal, essential, perfect, total, unique [...] (p. 531)

[The authors go on to provide counterexamples for many of these so-called absolute adjectives.]

"Gradability itself is not an all-or-nothing matter. Even conservative [grammar] manuals accept that adjectives such as complete, perfect, total, unique admit the degree adverbs almost and nearly [....] But these are matters of semantic compatibility, not of grammaticality." (p. 532)

All we can say is that although many adjectives are not gradable in the literal sense, they are sometimes used as gradable adjectives in special circumstances.

Marilyn

*Huddleston and Pullum (2002)