a and b are elements of the real numbers so I think your first statement was correct. If we say that R tells how the probabilities combine to get the expecation value then I think that R^2 will tell us about how the squares of the probabilites will add which tells us about the dispersion of the...
you don't have to write the plus explicity. |z>=|+z>=(1,0) which is not equal to -|-z>=(0,-1). The negatives are part of the nomenclature and do not have algebraic signifigance.
No since |z>=(1,0) in the z basis and |-z>= (0,1) in the z basis you could a. never get a scalar under addition and you could not get an answer of the zero vector since these vectors are linearly independent and form a complete basis. For your previous post you need to calculate what |-z> is in...