If z is a "standard" normal random variable, that means whatever its mean was, it has been standardized to ZERO, and thus the z-score table for the normal distribution is what you are meant to use here. Therefore the probability thatZ is less than 1.2 is equal to the cummulative probability at z = 1.2 on a z-score chart.

For the second one, you are told that the probability between -z, and z is 0.5. Do you understand what this means? Look at the values of "z"; they are equal correct, in the sense that z is the number of standard deviations away from the mean - -z simply goes to the left of Z, and +z goes to the right of Z. The easiest way to find z's is to fix Z as the mean (theres no way of knowing this because the information you post doesn't give us any indicator). Looking at a normal chart it looks like 0.65 is a good choice for a portion of the graph that is .25 percentage points; since by symmetry the other z needs to have the same amount of its brother - .5/2).