Trying to understand how a guitar pickup switch works

The switch you have is almost exactly like the one in the top diagram, the only difference is there are two contacts for each position, plus it has a pair of contacts for the "off" position which you can also ignore. So you can just wire it up as follows:

where the pin numbers 3, 4, and 5 correspond to the 3, 4, and 5 I have added to the circuit below:

Here is the circuit for the volume and tone control for each diagram:

Although they look different in your two diagrams, they're not; the only difference is the tone control on the bottom has the middle (wiper) and end pins reversed, which makes no difference electrically.

The volume control works as a voltage divider.

The yellow component is a capacitor, and yes it goes where the 1 µF is shown in the upper diagram (which I circled in red). Along with the resistance of the pot, it forms the bass tone control by creating a variable low-pass filter like the one on this page (under the section "Knobs and Controls"), which also has a lot of other guitar circuits you might be interested in.