Two different irrationals can't make a rational...
Yes, √3 + √5 is irrational.
It is irrational. Any number that cannot be written as a fraction is irrational. So if the Golden Ratio were rational, instead of a never-ending decimal number, you'd see a fraction. The official measurement is (1+sqrt5)/2. sqrt5 is irrational.
In general, no. It is possible though. (2pi)/pi is rational. pi2/pi is irrational. The ratio of two rationals numbers is always rational and the ratio of a rational and an irrational is always irrational.
4 + 2sqrt3
Yes. It is equal to (1 + sqrt5) / 2 = ~1.618 which, though irrational, is a purely real number; i.e. it has no imaginary component.
No. A rational plus an irrational is always an irrational.
[-1+sqrt(3)]1/4
No. The sum of an irrational number and any other [real] number is irrational.
Yes. In fact, a rational plus or minus an irrational will always be irrational.
Yes
Yes.
Yes, always.