Compute emitted importance of a pinhole camera

In section 16.1.1 of Physically Based Rendering the authors describe how we can check, if a given ray $$r$$ corresponds to one starting from the film area.

They implement this check such that it works for cameras with finite and pinhole apertures. They assume that $$r$$ has origin $$p$$ and direction $$\omega$$. Maybe I didn't get it, but are they assuming that $$p$$ is a point on the lens? That seems to be the case from Figure 16.1.

Now they want to find the point on the film that $$r$$ corresponds to. Let $$w$$ denote the camera viewing direction $$(0,0,1)$$ in world coordinates. Then they compute the cosine of the angle $$\theta$$ between $$\omega$$ and $$w$$, $$\cos(\theta)=\langle\omega,w\rangle.$$ Assuming we're dealing with a pinhole aperture, why do we find the point on the film by intersecting the ray with a plane arbitrary set at $$z=1$$? And why is this intersection at $$p+t\omega$$, where $$t=1/\cos(\theta)$$? Shouldn't the intersection be at $$t=\frac{\langle w-p,w\rangle}{\langle\omega,w\rangle}$$?

And anyway, doesn't $$p$$ need to be equal to the eye point of the camera?