Wow, I did not expect this many replies that quickly, thanks a lot!
What do you have on there for decoupling capacitors?
There is 100µF at the voltage regulator input and 10µF at the output. The MCU has 0.1µF, and according to the evaluation kit datasheet the MAX3421E has 2.2µF + 1µF.
Check decoupling. Add a series ferrite bead to the wire or reset line.
I'll have to buy some to test that in series. I just tried loosely putting a big clip-on ferrite bead around the reset wire and it made no noticable difference.
Aside from the boilerplate "this is a breadboard, it'll never work since it is a breadboard" sentence,
There are IC's that have very sensitive RST inputs. The Maxim chip might be one of these, so adding a small capacitor (1nF ... 10nF) from the RST pin to either GND or the supply voltage (depending on the active level of the input) usually helps. Place this capacitor really close to the IC in your layout (as I said - breadboards don't work well with fast digital stuff).
Indeed, I did run into problems with random corrupted data when talking to the MAX3421E through SPI @ 12MHz. I thought the breadboard capacitance(?) might be smoothing out the signals so I tried lowering the clock speed and the corruption went away. Full-speed USB communication between the PIC and the PC does work though (but there could be retried packets due to occasional corruption, I don't know).
Would using a soldered prototype board really be much better in this regard?
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Putting a 10nF capacitor closer helps with the resets! I'm holding it (will probably solder it on the wires later) right in front of the evaluation board's big connector and I don't get spurious resets when making contact with random metallic objects anymore (I had only tried putting such a capacitor on the breadboard side of the reset wire before). The MAX3421E's RES is active low. What is the best place to put the capacitor? I don't notice a difference between +3.3/RES or RES/GND.
However, it still resets if I'm connected to earth ground through the PC and I make contact with a device's USB receptacle while it's plugged in (or directly to it's barrel jack's negative source, which is not earth grounded). So the issue is not completely solved, but that's progress!
Is the pc where all the stuff is attached to grounded?
Yes: 11 ohms between the PC usb plug's outside and the earth ground pin of the PC's wall plug.
Why does that animation look so well timed after after each increment to 1?
It's 2X real-time speed to make the video nicer.