Yes. sqrt(2) + sqrt(2) = 2*sqrt(2), an irrational number.
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Yes - if I had an irrational number x, and I added that to the number (7-x), I would end up with 7.If the number is irrational, it can be subtracted from a rational/integer to make another irrational.
You can add any irrational number.
No, but you can add an irrational number and a rational number to give an irrational.For example, 1 + pi is irrational.
No. The set of rational numbers is closed under addition (and multiplication).