Yes, when you add any group of natural numbers, the sum will also be a natural number.
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Natural numbers are actually closed under addition. If you add any two if them, the result will always be another natural number.
Yes. The entire set of natural numbers is closed under addition (but not subtraction). So are the even numbers (but not the odd numbers), the multiples of 3, of 4, etc.
Natural (ℕ), integer (ℤ), rational (ℚ), real (ℝ) and complex (ℂ) numbers are all closed under addition.
Quite simply, they are closed under addition. No "when".
No. A number cannot be closed under addition: only a set can be closed. The set of rational numbers is closed under addition.