1. There are many people in the park.

2. There are many peoples in the park.
(Is #2 right?)

3. How many people are there in your family?

4. How many members are there in your family?

5. How many are there in your family?

(Are the three sentences above all correct?)

6. Do you want some cheese?
Do you want a piece of cheese?
Do you want a slice of cheese?
(Are all grammatical?)

2 answers

2. There are many peoples in the park.

That is a possibly correct sentence but it would mean that there were many different types of people in the park, such as Americans and Chinese and Solomon Islanders.

3. How many people are there in your family?

That is awkward but understandable. There is no need for the word "there" in the sentences.

4. How many members of the Society of Jumping Rabbits? How many members of what?

5. How many whats in your family? Horses?

6. Looks good to me.
1 and 2 can both be correct (as Damon explained about #2), but 1 is the more commonly used.

3, 4, and 5 are all correct, but (as Damon says) removing the word "there" would make them better. 5 is the best of the three.

6 - All three are correct.