If you mean that the number continues by a number of 0s which is one more than the previous numbers of 0s followed by a 2 forever, then it is an irrational number.
If you mean that the "2020020002" repeats then it is a rational number.
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Yes.
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It must be a generalised rational number. Otherwise, if you select a rational number to multiply, then you will only prove it for that number.
Yes. Any rational number divided by another rational number is also rational.