Is there any physics "gotcha" I need to watch out for with such a setup? For example, would it be possible to make a perpetual motion machine somehow by putting a demon on a properly positioned Ferris wheel or something?
Nope, you're good
This is not much different from magnets and ferromagnetic metals - supposing you have a huge magnet or an Oscar Isaac-amped Michael Fassbender at the south pole.
We can't do it with magnets, we won't be able to do it with demons.
Now of course, being close enough it would seem that your demon core would work like a magnetic monopole, since it is a point source. Still it wouldn't allow for perpetual machines. I saw someone commenting somewhere else in the internet that with a monopole you could have a DC motor without a commutator, which is an interesting application, but still has nothing to do with perpetual machines.
What you need to watch out for is that there might be a huge area of exclusion for demons. It is exact as you mention for one approaching the south pole, but it would extend far beyond it. A round'ish, not so dense demon left on the equator might reach the south pole in finite time by a combination of rolling and floating, and the closer they get, the faster they'll go and the harder it'll be to reverse their motion.
Edit: it just came, this opens up a new branch of physics called
I wish I could say this is not rocket science, but it is. I'll spare you the math but any demon accelerating towards the demon core from hundreds of kilometers or miles away and bypassing it - or being deflected by it - will go really really far past the horizon, possibly exiting the atmosphere.
Those who don't hit anything along the way will either eventually fall back and impact against the ground, possibly restarting the cycle, or given enough extra push from other sources will escape the planet (actually the core) altogether.
Those over escape velocity who hit something on the way after going past the demon core may be slowed down to orbital velocity. They will orbit the demon core rather than the planet's center of mass, and this is where things get interesting and bizarre.
Regular satellites will always cross the plane which contains their parent body's equator. This is not the planet for your demons, so if they have zero or little north-south movement when they start orbiting they will circle around the south pole in space, but never going too far away from it. This is just not possible for a satellite in real life and may have interesting applications. In real life when we want something over the poles we need a polar orbit, which means always visiting both poles in an orbital period, which is wasteful if you are interested in only one.
If their orbit has a considerable angle relative to the equator, though... If is it much larger than the planet they may be safe, but most likely it will intersect the planet (that is, it will be a pagh that passes through the planet). This means they will eventually hit the ground at bat out of hell speeds. We're talking about multiple miles or kilometers per second at the very least. Whether they bounce back, become a huge puddle or something else entirely is up to you.
With a caveat though
This is considering ideal conditions. Most likely, since your demons interact with matter, they will be slowed down by air. They will eventually reach a terminal velocity orders of magnitude less than escape velocity. The visible effect would be demons skidding towards the south pole at speeds comparable to that of an airplane. It would still hurt a lot to stay on their path.
Going further and beyond
I specially love this comment from Shufflepants:
I however imagine, because only demons are attracted by the demon core, there might be a potential for reducing the fuel cost of launching regular satellites into orbit by including some demons inside your launch craft, accelerating up and towards the south pole, and then jettisoning the demons as you pass over the south pole.