No
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The product of 0 and an irrational is 0 (a rational), the product of a non-zero rational and any irrational is always irrational.
The question cannot be answered because it is based on a false premise.The product of a (not an!) rational number and an irrational number need not be irrational. For eample, the product ofthe rational number, 0, and the irrational number, pi, is 0. The product is rational, not irrational!
Yes, unless the rational number is 0.
No. It is not defined if the rational number happens to be 0.
The product of a rational and irrational number can be rational if the rational is 0. Otherwise it is always irrational.