So let’s just say “as close as possible”.
If you mean the Pioneer receiver, each channel has a 10 band EQ. However, when you set the speakers to small, the 32 Hz band is not available anymore for obvious reasons. The subwoofer channel has a 4 band EQ. It’s a pain to setup each channel separately, but I’m used to it.
Sorry, not sure what you mean by “control system”.
Sorry, “control system” was ambiguous, what I meant was a reference system like your car audio, cheap computer speakers, etc.
If your mix sounds good on different reference systems, chances are higher that it will translate on other systems as well. I would keep it as is and not try to mess with it. Besides, it will be very difficult to adjust your system with these EQs.
But you do you. That’s what I would do.
And I can only repeat myself: Do yourself a favour and treat your room… every little helps. It’s literally sound advice 
Any tips on the best mic and software to test a listening/mixing/mastering room?
What kind of results should one shoot for?
There are probably good books out there on the topic but I’m oblivious to what they are and how to best hunt them down.
I gather software exists that you can play different kinds of test tones and pink/white nose and stuff…take shots of what’s happening where your ears usually are. No clue how to pick it, and set it up, and interpret the data.
Can that kind of equipment be ‘rented’ for a few days, as opposed to an outright purchase? I’m thinking the right mics might be rather expensive?
A popular software is Room EQ Wizard, at least in home theater and say semi-pro studios. They list a couple of mics they directly support, popular ones are the miniDSP UMIK-1 which is USB or the Dayton EMM-6 for which you’d need an audio interface. These mics usually also come with calibration files.
You’ll be interested not only in the frequency response as such, but also how it varies over time. And, in relation to that, also the reverbation time of the room. Which in a way is the same thing, just from another perspective so to speak.
There’s more high end stuff by companies like Brüel & Kjær, but then we’re talking another level of prices…
What @steve77 said.
In addition, here’s a calculator for room modes which gives you a first idea what you are looking for in terms of problematic frequencies and why you should do something about it.
Used mics with calibration files can be purchased online for a reasonable price, e.g. Sonarwork mics.
There’s plenty of literature out there and it is quite difficult to navigate to the good parts. The more you look into it the better an idea you get.
If you are going for a DIY solution then simple broadband absorbers are usually the most effective and easy-to-handle measures. It is of the essence to think about their placement and measures/material beforehand, though.
Room treatment makes all the difference. It is very rewarding and worth it. And more important than indirect purple light…