e.g. Do you want to know more about him?

(What is the part of speech of 'more'? Is 'more' a noun or an adverb?

1. Do you want to know know much about him?

2. Do you want to know a lot about him?

3. Do you want to know a lot of things about him?

4. Do you want to know many about him?

(Are the four sentences grammatical?)

In that sentence, "more" is a noun.

All of those sentences are grammatical except # 4.

The word "more" in your example sentence "Do you want to know more about him?" is functioning as an adverb modifying the verb "know." It is indicating the extent or degree to which you want to know about him.

To determine whether the four sentences you provided are grammatically correct, we can analyze each one individually:

1. "Do you want to know know much about him?" - This sentence is not grammatically correct. The word "know" is repeated, which is incorrect. It should be rephrased as "Do you want to know much about him?"

2. "Do you want to know a lot about him?" - This sentence is grammatically correct. "A lot" is functioning as an adverbial phrase modifying the verb "know."

3. "Do you want to know a lot of things about him?" - This sentence is grammatically correct. "A lot of things" is functioning as an adverbial phrase modifying the verb "know."

4. "Do you want to many about him?" - This sentence is not grammatically correct. The word "many" is being used as an adjective, but it should be modified by a noun. It should be rephrased as "Do you want to know many things about him?" or "Do you want to know many details about him?"

Remember, when analyzing the grammaticality of a sentence, it's important to consider the correct word usage, word order, and proper sentence structure.