Yes.
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1, 3, 7, 9, 21, 27, 43, 63, 129, 189, 301, 387, 903, 1161, 2709, 8127.
A prime number is a number with two and only two factors: 1 (also known as unity) and itself. 8127 has factors other than 1 and itself, so it is not a prime number. For example, it is divisible by 3. (You can check this by adding the digits of the number: 8 + 1 + 2 + 7 = 18. If the sum of the digits is divisible by 3, the number is also divisible by 3.) Therefore, 8127 is a composite number.
Yes; all numbers that are divisible by 9 are divisible by 3.
Yes, 5553 is divisible by 9.
Yes, 682 is not divisible by 9.