MIMF

The previous owner of this guitar installed an undersaddle pickup. Supposedly he had a professional do it, but after looking inside I'd say that's BS - he did it himself & not very well, either. The adhesive foil was applied too far towards the bridge, so in its current state: 3 or 4 of the tuning pegs have punctured through, and on the treble side the foil is folded up against them, which is a bit of a PITA when changing strings.

So do I trim the foil back (at the treble pegs? all the way across?), paste it down and puncture the last couple pegs through, or leave it alone? I'm not really clear on the construction of piezo pickups, do I don't know if the foil is integral or just an adhesive.

In one of the pictures you can see that there's a little disc epoxied(?) to the soundboard, as well. What on earth is that, something related to the pickup?

That foil appears to me to be some form of electro-magnetic shielding, not the pickup itself. The pickup is usually located under the saddle but on top of the bridge so accessible only from the outside of the guitar. All piezo saddle pickups I've ever seen already have their shielding integrated. That disc might have been an early piezo pickup that has since been disconnected. It could also be some bizarre form of mute to tune the guitar's response? If the pickup works correctly and the string response is even then I wouldn't mess with it. If that foil is giving you fits then you could gently peel it away from the 2-3 treble pins unless someone else says otherwise.