Do I use an ICC printer profile within Photoshop or Lightroom ("PS manages colors”) when I print with the QTR printer?

Do all your printing through PrintTool so you can have more control over printing, to bypass color management, and make it easier to place the image(s) on the page. Printing through Photoshop might be what is adding to your confusion and difficulty in getting started.

Is a custom paper and ink ICC profile indeed solely used for on-screen soft-proofing?

Yes, once you build the calibrated media setting, the ICC profile is mostly or soft-proofing (and honestly, I never soft-proof or print using color management because I trust what my calibration software does and the matte ICCs can often over-compress the shadows)

Do you have a Windows version or can it be run on a Windows computer in macOS virtual machine?

I’m sorry, I no longer develop or support the windows version of the software. QTR on windows isn’t the best way to do things because of the limits of the QTR gui and weird windows security stuff. 
Microsoft also makes things too hard and expensive for a solo non-windows focused developer to do the kinds of things I need to with these apps (and I would barely call myself a “developer” at that.) 
I highly encourage everyone to find a used Mac mini or MacBook Pro from 2015 or later to use only for printing and calibrating with QTR. It will make you life 10000x easier.

What are the differences between the QuadToneProfiler K3 and Pro apps?

The QTP-K3 app only has starter CMYK curves for a single gloss and matte paper with a neutral base (Canson Platine and Edition Etching) and the basic linearization tools.

The QTP-Pro app has the same Starter Curve Setup, and more advanced linearization tools for the gray and color inks. The main difference and benefit of the Pro app is the custom curve creation tools for printers or ink sets with up to 8 gray inks. Even if you only have the OEM Epson inks, it has the ability to create custom profiles for different papers.

The most powerful part of the Pro app is the ability to blend the base gray inks with the toning inks to create a special .QTP file that has the quad curves and measurements for up to 4 individual gray or toning curve components that can be blended to preview the toned color in real-time. You can also save the custom toner presets into the .qtp file. The Pro app also has additional tools for editing, reshaping, applying curves and levels adjustments the .quad files, ink channel remapping.

What are the differences between the QuadToneProfiler-Pro and Pro-DN apps?

The QTP-Pro and Pro-DN apps do have some overlap for the additional tools like the blending, levels, quad editor, and remapping tools, but the negative/positive correction math and the built-in starter curves are very different. You will need both apps depending on what you want to do.

QuadToneProfiler-Pro is designed around inkjet positives and has specific tools for creating customized multi-gray ink sets with additional color toning inks. It can be used for photogravure printing, but the main use is for creating custom curves and blending the toning inks to preview the output color of the black and white prints for different amounts of gray+toning inks.

The QuadToneProfiler-Pro DN is only for inkjet negatives for creating curves with customized ink sets or for processes that do not work well with the built-in K3 starter curves.

I tried to install a new quad file with the install printer command file and am getting an error that I don't have permissions?

You need to reinstall QuadToneRIP from the downloadable install package from, http://www.quadtonerip.com/html/QTRdownload.html .

The cause of (nearly) all permissions problems is macOS updating and it reverting all the permissions that QuadToneRIP sets for the library/printers folders.

After reinstalling QTR, run the install command for your printer(s), and then quit and relaunch PrintTool and it will all be solved.

Does QTR indeed use solely the/a black ink cartridge and only add "a dab" of color to create the non-neutral (cool, warm, sepia) prints?

QTR gives you total control of which ink and how much ink to print so you could print black ink only, or print red ink only if you wanted. It is all based on what you build into the .quad. QTR will simply take the luminance values from the original image and spit out as much or as little ink for any channel in the printer.

If you print with only the MK or PK ink in the OEM inkset, you will probably see some dots/graininess in the midtones and highlights and a color cast. You will need to adjust that by adding a little cyan and magenta to balance the warmth of the black ink. I would really recommend ditching at least the orange and red inks for two extra dilutions of gray so you can have smoother prints and then just use the CMY inks for toning. I built a suite of custom QTR calibration software to make really accurate toning curves that you might benefit from using—especially if you start to swap out inks.

I cannot select whether to use PK of MK ink in my specific printer. Is this automatically selected based on paper selection (gloss or matte)?

Printers that have dedicated slots/nozzles for MK and PK (that don't switch in the printer), use the values in .quad file to dictate what ink to use. Paper selection has nothing to do with it.