So, should I/we try to EQ the various instruments at the mixing stage to avoid any frequency overlap whatsoever? That even possible? Would using a spectral analysier help identify the exact frequency range of an instrument and thus help with identifying which frequencies to cut / boost?
The short answers: No. No. And no.
As the author/creator of one of the most popular frequency charts on the Internet, which is also the single most popular feature of my website (click on my signature logo), I have to tell you that those charts are completely misused and abused, and are next to useless for the purposes most folks actually want to use them for, your questions being a perfect example.
There's one fact that sticks out like a sore thumb in those charts, the very first thing that someone should notice when looking at them, but almost nobody does: those instrument frequency ranges are *almost all* overlap. If there's only one lesson anyone should get from those charts, that should be it; because that knowledge leads to the answers to so many questions, including two of the three you asked.
It is impossible to remove overlap in instrument frequency ranges. Except at the extremes (bass vs. piccolo), there is a potential for overlap almost everywhere.
This is also one (of many) reasons why you can't just simply look at a spectral analyzer and carve out spaces for each instrument that way. Spectral analyzers are a good way for trained ears and eyes to help identify a limited list of specific ailments within a mix, but simply cannot and should not ever be used as a way of creating a good-sounding mix. Illnesses like sibilance, harmonic distortion, mp3 encoding artifacts, etc. can help be identified and quantitized using an RTA, but there is no way by looking at an RTA to determine if something sounds right or wrong, good or bad.
Instead, tiger, 95% of the "roles" you select for each instrument should be taken care of in the music composition and arrangement itself, with your mixing plan set to support that arrangement.
For the remaining 5%, when you have instruments are still competing and conflicting with each other, you have two main tools: first, you can separate them in the pan space. Second, you can run a little differential EQ between the two of them; e.g. select one to be a little more low end heavy and the other to be a little more high end heavy, perhaps, and then by just a couple of dB each, boost one where you cut the other in order to slightly shift the tonal balance a bit. They will still greatly overlap, but the emphasis will be just different enough to keep them from sounding the same.
G.