as + adjective + as vs as + many + as Rules

Some of your examples are correct but your rules around "as ... as" are not correct. Here are the rules about how to use "as ... as".

1. Use "as + adjective/adverb + as" to make comparisons when the things we are comparing are equal in some way:
   • The jewels are as beautiful as the ocean.
   • The world’s biggest bull is as big as a small elephant.
   • You have to unwrap the present as carefully as you can. The present is delicate.
   • They played as well as they usually do.

When we use "as + adjective + as" we are comparing the qualities of nouns. For example, the first example is comparing the beauty of the jewels to the the beauty of the ocean and saying that they are equaling beautiful.

When we use "as + adverb + as" we are comparing the qualities of verbs. For example, the third example is comparing how carefully you HAVE to unwrap the present to how carefully you CAN unwrap the present and saying that they have to be equal because the present is delicate.

2. Use "not as ... as" to make comparisons between things which aren’t equal:
   • The books are not as heavy as the ball.
   • Rory has not grown as tall as Tommy yet.
   • She is not singing as loudly as she can.
   • They did not play as well as they usually do.

Like rule 1, when we use "not + as + adjective + as" we are comparing the qualities of nouns. For example, the first example is comparing the weight of the books to the weight of the ball and saying that they are not equal.

When we use "not + verb + as + adverb + as" we are comparing the qualities of verbs. For example, the third example is comparing how loudly she SANG to how loudly she CAN SING and saying that they are not equal. The trick when using "not as ... as" with verbs and adverbs is that the verb has to come between the not and the as before the adverb like "not + verb + as + adverb + as".

3. Use "as + quantifier + noun+ as" to make comparisons about the quantity of two things:
   • Greg makes as much money as Mick.
   • They try to give them as much freedom as they can.
   • There weren’t as many people as I expected at the party.

The first example is comparing the quantity of money that Greg makes to the quantity of money that Mick makes and is saying that they are equal. The third example is comparing the quantity of people at the party to the quantity of people that I expected at the party and is saying they are not equal.

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I've rewritten your examples so that they are correct and state which rule is applicable to each:

• The jewels are as beautiful as the ocean. -> The jewels are as beautiful as the ocean. (rule 1)
• The jewel as beautiful as the ocean -> The jewel is as beautiful as the ocean. (rule 1) Note: you forgot the verb.
• As beautiful a jewel as the ocean. -> The jewel is as beautiful as the ocean. (rule 1) Note: you cannot order the sentence like "as + adjective + noun + as + noun"
• There are as many dogs as cats. -> There are as many dogs as cats. (rule 3)
• A risk as little as one chance in a million -> The risk is as little as one in a million. (rule 1) Note: you forgot the verb and the idiom is one in a million or one in a million chance.

For more information see: https://dictionary.cambridge.org/grammar/british-grammar/as-as