An irrational number is a number that has no definite end. So it can't be multiplied or divided by anything to make a rational number that does have a definite end.
It can be a rational number or an irrational number. For example, sqrt(2)*sqrt(50) = 10 is rational. sqrt(2)*sqrt(51) = sqrt(102) is irrational.
The sum is irrational.
No. If the rational number is not zero, then such a product is irrational.
Yes, it will always be irrational.
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 product of two irrational numbers may be rational or irrational. For example, sqrt(2) is irrational, and sqrt(2)*sqrt(2) = 2, a rational number. On the other hand, (2^(1/4)) * (2^(1/4)) = 2^(1/2) = sqrt(2), so here two irrational numbers multiply to give an irrational number.