AFAIK, the two main factors are string construction and the location of vibrational nodes relative to the pickup. The former: thinner strings are thinner, less rod-like and have different relationships between the harmonics. The second means that for an arbitrary note played on different strings, the pickups also "see" slightly different relationships between the harmonics that are present *. Unless you have EQ for each string that also changes with each note you play, EQ can't fix the sound entirely. Boosting EQ at 98 Hz won't help if I'm up at the 9th fret on the G and won't help the second harmonic of the open string.
A D string fretted at the 5th fret is a G but sounds different to an open G, or if you want to take the open string out of the equation, then the Ab.
I think about the best you might get a strings that are constructed, maybe different core and wrap ratios, to minimise the harmonic differences, but I expect it would be hard to do any maintain consistent tension and elasticity across a set of strings.
* I had one electric guitar that if I selected one pickup and picked a harmonic then at one of the standard locations - 5th or 7th fret - it was entirely silent. It's a very narrow set of harmonics when doing this and the single coil pick was at a node.
OK, I'll go through this one thing at a time:
1) The harmonics on your G string and those on all your other strings are very much the same in terms of where they are relative to the fundamental - it
isn't the string construction that makes the G sound weak on a 'Ray - if it was, it would infect every other electric bass type, and....we don't hear that. The difference in string construction also has nothing to do with getting more or less low end (what we're talking about here) - the bigger strings have more windings to try to keep the high frequency harmonics in line (and they do a good job at that - at least for round wounds - flats lose the harmonics so quickly that the fact that they are a bit off doesn't really matter much).
2) You are right that pickup location is important. If you want to fix the low frequency deficiency caused by pickup position (The 'Ray needs this as the pickup is much closer to the bridge), the EQ does need to change for each string to be "correct", but it doesn't need to change for each note you play on a string - the loss of low end is the same for every note on a string in a frequency response sense. As you go up the string, you don't need as much boost to the fundamental, because the fundamental is higher in frequency - a fixed low boost filter (which boosts the fundamentals of the lower notes more than those up the string) would correct the loss of low frequency energy for every note
on that string.
Here's a response curve of what happens (on one of my basses) for all 5 strings - each string has it's own curve. There's a lot going on here, and this isn't a 'Ray, A 'Ray has more rolloff on every string due to where the pickup is) but it shows the concept:
What you see is that, at low frequencies, every string has a different rolloff. The G string has the most rolloff, so it needs the most help, but with one filter in the preamp (tuned to help out the lower strings), it doesn't get enough help - it sounds "weak".
3) I've been meaning to do a post explaining the Physics of how pickup locations affect basses (like my post on tone controls), and I have some of the work done, but it's a somewhat involved topic, and figuring out how to present it all in a manner where most folks can understand this is not a quick task - I will get it posted, but it may be a while.
The harmonics thing you mention - it's there on every string instrument. A 51 P bass has a very prominent one - the pickup is at 6.8 inches from the bridge, which is 20% of the string length, so yes, one of the harmonics - if you play it, you can hear it acoustically, but in the electrical output it is not there. This is part of the comb filter response, which is really obvious at a particular frequency - the notches in a comb filter act like that, but the low frequency rolloff of the comb filter is much broader - it affect a lot of notes, but not as much as a notch - it attenuates fundamentals, but they don't go completely away.