Natural numbers are non-negative whole numbers (no decimals or fractions). This would be 0, 1, 2, 3, 4, 5, 6, etc...
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Yes.
There is some disagreement. Some people include zero in the set of natural numbers (like whole numbers), some people don't (like counting numbers).
It is the set of natural numbers.
Please clarify what set you are talking about. There are several sets of numbers. Also, "closed under..." should be followed by an operation; "natural" is not an operation.
No - the ONLY natural number which has a reciprocal which is also a natural number is "1". All the other reciprocals of natural numbers will be fractions that are less than 1.