yes, 3 is a natural number. any positive number is.
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Negative numbers are not natural, but there are negative integers. Examples are -1,-2,-3,-4, and so on. These are all integers but none of them is a natural number.
Pi is not a natural number. Natural numbers are just the regular set of numbers used for counting (1, 2, 3, 4, ...). Pi, being not even a whole number, is not natural.
Sometimes. A rational number is any number that can be written in the form p/q where p and q are integers but q not = 0. So 3 is a natural number and a rational number because it can be written as 3/1. But 1/3 is a rational number only because it will not reduce to a natural (whole) number.
Sum of two rational numbers might be a natural number (1/2 + 1/2), but mostly it's just another rational number (1/2 + 1/3). So answer is no.
The product of a natural number and all of the lesser natural numbers is called the factorial of the number. It is often represented with an exclamation point. So:5! = 5 * 4 * 3 * 2 * 1