How should I think about sunlight for the purposes of analyzing the angles it makes w/ earth? Most diagrams I see of sunlight look like something like this:

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They show that for basic geometric analyses, the sun's radiation can be thought of as 'thin lines' that strike earth. Further, at an equinox, those rays strike hit the horizon of every point on earth (except the poles) directly from the east. So, the diameter for the earth does not matter at all it terms of the angle at which the rays strike.

I'm curious - how far will this conceptual frame take me? What is a better way to conceptualize how the sun's light strikes the earth?
This is a first-order approximation, and works well for simple models of irradiation of the atmosphere or the Earth's surface.

But, a simple experiment with a pinhole camera or a single lens will show that this model doesn't work in imaging situations.  The sun is an "extended object," meaning it is not a point source, so the image you get is not what you would expect from a point source.  Worse, even if the sun were a point source, the distance to the geometric focal point is different very slightly from the distance if all the sun's rays were perfectly parallel.  See the Lensmaker's Formula,  $ \frac{1}{f} = \frac{1}{p} + \frac{1}{q} $ where p and q are the distances from the lens to the source and the image respectively.   (I'll warn that for a reasonable lens, say focal length $ f = 10 m$, the difference between focal location for rays that are parallel and rays from a point source $150\space million\space km$ away is ridiculously small) 

Similarly, if you want to model the umbra and penumbra during a solar or lunar eclipse, you'll need to take the physical extent of the sun into consideration.