LTI system phase response given $z$-transform
Answer to (b):

$(1+z^{-2})$ is a Type I generalized linear phase filter with group delay 1 (i.e. the phase response is $\angle H(e^{j\theta})=-\theta$ with a jump of $\pi$ at $\theta = \pm \pi/2$).

6 of those filters in cascade will have group delay 6 (phase response $\angle H(e^{j\theta})=-6\theta$).