Is the book wrong? -- How to sketch frequency response obtained from H(z)?
As far as I can tell, the book is not wrong. (By the way, you're studying a 3-point moving average filter here, and it's useful to know about such filters.)

Your 1st question: That center dot on the Mag. and Phase curves serves no purpose that I can see. I suggest you not worry about that center dot.

Your 2nd question: The Phase curve is generated by evaluating the angle of the  $e^{-j\hat{\omega}}$ phase term, in the last line of the $H(\hat{\omega})$ expression in Part (c) of the Solution, as $\hat{\omega}$ goes from $-\pi$ to +$\pi$. As $\hat{\omega}$ goes from $-\pi$ to +$\pi$, the Solution's $\hat{\omega}/2\pi$ Phase curve freq axis label goes from -0.5 to +0.5.

(Note: as you study the Solution know that the $H(z)$ in Part (c) is not the $H(z)$ in Part (b). The $H(z)$ in Part (c) is the $H(z)$ from the Solution's Part (a).) 

Your 3rd question: It's a good question. In the literature of DSP there are a number of different notations (different symbols with different dimensions) used to represent frequency. And those various notations certainly can be confusing at first. mmswe, have a look at the following web page, it might help you with your 3rd question:

http://www.dsprelated.com/showarticle/199.php