When I click on something, it selects, I believe, the largest object in a given radius from the click offset, not the path I've actually clicked on. Is there a way to increase the precision? Zooming in and out every time is starting to become a tedious operation...
Normally Inkscape selects the topmost that occupies the exact position where you click.
There are many ways of selecting objects with the mouse. You may be interested in /select under/ for selecting objects not at the top.
From Dmitry Kirsanov's book:
Also, you will notice that every clickable object has an invisible margin, several pixels wide, on all sides. A click in that margin will still select that object. [...] On the downside, this also explains why it's sometimes so difficult to select the bottom object in a stack even if that bottom object protrudes a little from under the top one.
If you don't like the size of this clickable margin, you can change it on the Mouse tab of the Inkscape Preferences dialog: Adjust the Grab sensitivity value, the default being 8 screen pixels.
For some reason, I had to restart Inkscape for this to take effect.