"Single coil" and "Humbucker" aren't standards for frequency spectra. There are enough variables within each technology—everything from the type of magnet to the number of coil windings to the material and thickness of the coil wire—to make a huge amount of variation possible. And that takes none of the other very important variables of tone into consideration: the density of the guitar's wood, the type and thickness of the strings, the position of the pickup along the strings, the electronics inside the guitar, the closeness of the pickup to the strings, the playing technique, and probably lots of other stuff I can't think of off the top of my head.
Depending on the alternating current service in the country where the guitar is plugged in, the hum comes in at 50 Hz or 60 Hz. The lowest fundamental frequency of a guitar is 82.4 Hz (E2), so the brunt of the hum can be easily and safely removed (with a parametric EQ notched at the appropriate frequency) without fear of affecting the tone. (The waveform of alternating current in electric power circuits is almost always a sine wave, so its impact at other frequencies is theoretically nothing and, in practice, dependent on the local conditions.) As you know, that alone wouldn't make a "typical" single coil pickup sound like a "typical" humbucker. (A noise gate is nothing but a volume-triggered on/off switch. In theory, it does nothing whatsoever to change the sound.)
Assuming you could exactly establish the frequency response of a humbucking pickup, and each of those frequencies was sufficiently present in the signal from a single coil pickup, you could probably create an EQ curve which makes the waveforms look alike in a Fourier analysis. But that's just for one note played at one loudness.
As you alluded to, the frequency response changes with the amplitude. So you'd need a compressor to mimic the exact behavior of the humbucking pickup or a sophisticated dynamic EQ.
But even then, it wouldn't be identical. Even if you have the same player, guitar, pick, pickup, strings, amplitude, and note, you wouldn't ever get the exact same thing twice. There are way too many variables at play.