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There are 2 main rules in assigning frames following DH convention: source: Robot Modeling and Control, Spong et. al $x_i \perp z_{i-1}$ $x_i$ intersects $z_{i-1}$ In your attempts, Your first attempt is incorrect as your $z_2$ axis is not in the direction of actuation of the prismatic joint. I'm not clear on your second attempt as $x_1$ appears to be ...


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Kinematics is based on geometry and joint constraints, so no change there at all. Dynamic motion results from the sum of forces, so the only change to a dynamic analysis is that you don't include gravity, but all other forces would remain.


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Seems that the book has an error. You are right, they should correct that. In the drawing it can be seen that those two z-axis are indeed parallel. I recommend that you considered them parallel and try to draw them to see that it fits with the image.


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