Yes, except for the number 1, which is a perfect square but not composite.
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81 is; Natural number, Perfect-square number, Perfect 4th root number, Perfect totient number, Odd Number, Composite Number, etc.
No, 16 is not a square of a composite number. It is a square of a prime number, which is 4.
31 is a prime number. Please note that all perfect squares are composite, except 0 and 1.
4, which is the square of the prime number 2, is the smallest composite number.
If by 'perfect number' you mean 'perfect SQUARE', then yes. 49 is an odd number and a perfect square.