1. He inquired of me a question.

= He inquired of a question of me.

(Are both the same and grammatical?)

3. He chose me a chestnut.
He chose a chestnut for me.

4. He found me a good library.
He found a good library for me.

5. He left me a persimmon.
He left a persimmon for me.

(Are the expressions correct from #3 to #5?)

6. The trees grew green.
The leaves turned yellow.
He felt asleep.
It is getting warmer and warmer.
He grew fat.
He became pale.
The fish went bad.

(There are intransitive verbs taking complements in the sentences above. Are the expressions all grammatical?)

1. He inquired of me a question.

= He inquired of a question of me.

(Are both the same and grammatical?)
No, they are not grammatical. "He asked me a question" is all you need to say.

3. He chose me a chestnut.
He chose a chestnut for me.
The second one is correct.

4. He found me a good library.
He found a good library for me.
Either one is correct.

5. He left me a persimmon.
He left a persimmon for me.
Either one is correct.

6. The trees grew green.
It'd be better to say "the trees became green."
The leaves turned yellow. Correct.
He felt asleep. I think you mean "he fell asleep" or maybe "he felt tired."
It is getting warmer and warmer.
Correct.
He grew fat.
Correct.
He became pale.
Correct.
The fish went bad.
Correct.