Those are good questions.
I have seen songs, Castlevania comes to mind, where they will add or subtract 1 pitch unit from one of the square channel (by setting a single value of -1 or + 1 in the using the "Pitch" effect track on the very first note of the song). This seem to break the interference that you mention and it creates very interesting effects. Channels become much more audible all of a sudden.
I didn't write the sound emutation for the base NES (i did write some of the expansions that were missing tho) but it should be fairly accurate. It will not be helpful at all for you, but "the phase of the square channels resets every time hi-byte of the 16-bit period is written". That was quite a mouthful. In plain English it means that the phase will only reset when you cross over these notes (approximately) : A-3, A-2, D-2, A-1, F-1, D-1. I dont think its realistic for you to purposely reset the phase in your songs...
I just have an Everdrive N8. The base version (need to get the pro soon) does not have an NSF player, so i use my ROM exporter. And no, i dont have any mods so if i were to use expansion audio, it would be emulated by the Everdrive... which would kind of defeat the purpose of using real hardware. I heard of people having boards with VRC6 and other expansions on them, but I dont know much about those. If you have a Famicom, getting an FDS would be super cool. That's the one expansion chip that anyone can get and run on actual hardware easily.
-Mat