Are there specific places where pickups need to be placed on electric guitars? Given that most have two or three, how are the optimum positions calculated? On Teles, the neck pup is exactly under an open string node - which to me seems odd, and the closer one gets to the bridge, the more chances are that nodes will be 'in the way'. Leo seems to have considered most things; did he have a rationale at the time, or are the positions purely arbitrary?
Short answer: No
Pickups can be placed anywhere. What will happen is the sound you get from a pickup is highly dependent on placement, as you observe, because of emphasis of some vibrational modes over others.
Note that the induction field of a pickup is not one dimensional, so even if you place a pickup at a node for a certain mode of vibration, the pickup will still pick up that vibrational mode, just with a lower amplitude than other placements.
Of course pickups placed closer to the end of a string will favor upper harmonics over the fundamental, and vice versa with pickups placed closer to the middle of the sounding length. That means when you fret a string, you change the harmonic balance that a given pickup detects. That's one (of many) reasons why the same note played on different places on the neck sounds slightly different.
Many other factors affect pickup sound, but placement is definitely a major component, if not the most important. That said, placement is entirely a matter of taste.