NP + "had better" + Infinitive V

Question

It just occurred to me that this construction is very peculiar.

Pronoun:      I had better get going.
NP:           The cat had better be home.
Expletive:    There had better be food on the table.
Gerundive C:  Eating cake had better be good.
Wh-C:         What he saw had better be worth seeing.
Infinitive C: ?To fish had better calm my nerves. 
Relative C:   *That she kissed the frog had better amuse me.

The subject has to be a NP or a pronoun and can't be a relative clause. This may be because relative clauses are abstract and can't be the things or actions someone would expect to have done or been something. Infinitive clauses should be grammatical, but I'm not so sure. The verb that follows "had better" being infinitive is, I suspect, due to the subjunctive.

I can't think of any other example where it would be grammatical to substitute the word "better" with any other comparative (adj/adv), noun, or verb ("better" can be a noun and infinitive verb).

Questions:

What part of speech is "better" in the construction?

Are there any other constructions quite like it?

Is this an example of a fossilized construction?

Answer 1

The OED has this under "have" 20 a:

The past Subjunctive had = would have, is used idiomatically with adjectives (or adverbs) in the comparative, as better, liefer, sooner, rather; in the superlative, as best, liefest; or in the positive with ‘as’, as good, as lief, as soon, as well, to express preference or comparative desirability.

In the earliest form of these expressions, in Old English the adjs. léofre, betre were construed with be and the dative, e.g. him wǽre betere = it would be better for him. In Middle English, side by side with this, appears have and the nominative, in the sense ‘he (I, etc.) would hold or find it better or preferable’. The use with the positive, and superlative, and the extension to rather are later; the use of as soon, sooner, well, is recent, since liefer and better began to be felt as adverbs. (See exhaustive treatment by F. Hall in Amer. Jrnl. Philol. II. 281.)

Answer 2

You'd better go home now (you'd is you had, past subjunctive).

This formula of expression is not logical, it is transformed and there is no convincing explanation how it came to this illogical form.

In normal grammatical English this would be:

It would be better for you to go home now.

But that's much longer. You simply have to see the above formula as a special idiomatic expression. And sometimes in the course of times such expressions change their form.

Obviously some speakers feel that something is wrong with this expression and an attempt is made to better it by dropping "had":

You better go home now.

"better" is the normal comparative of the adjective good.

In Oxford's explanation (have, 20a, posted by fdb) you get an idea that the above mentioned expression was transformed several times. But it would not be easy to reconstruct the various and exact stages of development of this expression.