Sure; for example, 10 + pi is irrational, 10 - pi is irrational. Both are positive. If you add them, you get 20.
Not necessarily. The sum of two irrational numbers can be rational or irrational.
Yes, you can.
Yes. The sum of two irrational numbers can be rational, or irrational.
You can not add irrational numbers. You can round off irrational numbers and then add them but in the process of rounding off the numbers, you make them rational. Then the sum becomes rational.
No, the result is always an irrational number. In more advanced math it is possible to add an infinite amount of rational numbers by way of Taylor Series and get an irrational number. This is how numbers like "Pi" and "e" are derived.
Positive numbers always result in positive numbers when added, divided, or multiplied by another positive number.