Such a sum is always rational.
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Can be rational or irrational.
Yes
Sure. For example, the sum of:root(2) and: - root(2) is zero, which is rational.
No. Two irrational numbers can be added to be rational. For example, 1/3 + 2/3 = 3/3. 1/3 and 2/3 are both irrational, but 3/3 = 1, which is rational.
In between any two rational numbers there is an irrational number. In between any two irrational numbers there is a rational number.