4: 2, 3, 5, and 7
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Yes. All prime numbers and composite numbers are positive integers, or whole counting numbers. That leaves infinitely many numbers that are neither prime nor composite. If you intended to narrow the scope of your question to the whole counting numbers or to the positive integers, then there are NO such numbers that are neither. A counting number, however large, will be either prime or composite.
9.
Any prime number is relatively prime to any other prime number.
wow. no its not a prime number , its even for a start meaning its not prime , and it has many factors , 1,2,4,8 ...
Square numbers have too many factors to be prime.