1. He has as many books as Tom.

2. He has as much money as Tom.
3. He is as strong as Tom.
4. He runs as fast as Tom.
5. He jumped rope more than Jane.
6. Tom jumped rope as many as Mike.

(Are they all grammatical? Would you check the sentences, please?)

All are fine except #6. You need to add a word: "Tom jumped rope as many times as Mike."

Yes, all six sentences are grammatically correct. Here's an explanation for each sentence:

1. He has as many books as Tom.
This sentence compares "he" and "Tom" in terms of the number of books they have. To check whether it is grammatically correct, ask yourself if the subject ("he") and the object/comparison ("Tom") agree in terms of quantity or number. In this case, since both "he" and "Tom" refer to people, it is grammatical.

2. He has as much money as Tom.
In this sentence, "he" and "Tom" are being compared in terms of the amount of money they possess. Again, if you check for subject-object agreement in terms of quantity, you will find that this sentence is grammatical.

3. He is as strong as Tom.
The sentence compares "he" and "Tom" in terms of their strength. The subject ("he") and the object/comparison ("Tom") agree, so the sentence is grammatically correct.

4. He runs as fast as Tom.
Here, "he" and "Tom" are being compared in terms of their running speed. The sentence is grammatically correct as the subject and the object/comparison agree.

5. He jumped rope more than Jane.
In this sentence, "he" is being compared to "Jane" in terms of the number of times they jumped rope. Since the subject and the object/comparison agree, the sentence is grammatically correct.

6. Tom jumped rope as many as Mike.
This sentence compares Tom's and Mike's jumping rope ability or number of jumps. Once again, subject-object agreement in terms of quantity is maintained, making the sentence grammatically correct.

In summary, all the given sentences are grammatically correct and compare two people in different contexts.