Warm but not hot.
I will restrict myself to commenting on QTR. It assumes nothing. On Windows it is a stand alone application that is not color managed and whatever color profile is or isn't embedded in the image file will make absolutely no difference to how QTR prints the image. It's up to you to manage the ICC - assign or convert (as per our other thread) - before you save the file and send it to QTRGui. You can send an untagged image, or one where you've assigned any profile you like, to QTRGui and you'll get the same print. The one thing that
will make a difference is if you convert to another ICC, because then for an untagged file, Photoshop will assume your default grayscale working space and convert the file, changing it. That's going to change the print, but not because QTR did it, Photoshop is the culprit.
On OS X, QTR is a printer driver and it too is color profile agnostic. It's just a relatively basic printer driver. It doesn't expect anything. The printing software is another matter. Both Photoshop and Print tool offer options to convert to an ICC on the way to QTR. Since OS X 10.6.8 you can't print untagged or gray gamma 2.2 images from Photoshop using QTR without getting a silent profile conversion behind the scenes. This was the precise reason that Roy Harrington created Print Tool - to allow you to do that. Yes, ACPU does as well, but Print Tool is more functional. So to get the same degree of control as in Windows, use Print Tool (or ACPU).
So QTR has no expectations. The question of what ICC to use is really about what workflow and software you are using. Most of the documented workflows assume gray gamma 2.2, so that is how you need to print the target. Your print the target the same way you are going to print images, taking care to avoid any silent profile conversions. If you print in Windows using QTRGui, or in OS X with Print Tool with CM turned off, you can print an unassigned target, or one with gray gamma 2.2 or AdobeRGB (same as gray gamma 2.2 but in RGB space), and it won't matter. My impression is that Roy Harrington supplied his test charts as untagged images because he assumed that people would be printing them that way, in line with generally accpeted workflows, but that's just an impression, and it's about workflow, not software.
My understanding of the ICC profiles that ship with QTR, like gray-lab.icc, is that they were created by Roy Harrington as generic profiles for people to use long before devices like the i1 became widespread and people started creating their own. They're mostly dated around 2005.
All images have a color space baked into it even if it has no tag or sign that says: Hello world, my creator made me to be viewed in the sRGB color space!
Yes and no. An image file is just a bunch of numbers and an ICC profile is a guide to color managed software as to how to interpret it. It's true that if someone sends me an untagged color image then I will want to know what color space they intended, because that will tell me how it is
supposed to be viewed. The same is true for a B&W image, but not so for test charts. No-one asks what color space the x-rite test charts are in, for the purposes of creating a color profile. And so it is for QTR under most workflows.