This has to be a long post. No way around it. I'll try to make it as straightforward as possible.
First of all, just to get a baseline, make absolutely sure you are using the correct paper profile. This is set in the Photoshop print dialog. Then you also need to go into the printer driver and pick the correct paper type there (this controls total ink). Turn off printer color management so you don't get double profiling.
Next, the principle and background of display calibration:
And it's actually very simple: you calibrate the screen to match the print. You can't adjust the paper, but you can adjust the screen. The paper color and brightness is a fixed reference. So is the maximum ink density, the deepest black. So that defines the white point, and the black point. This is the environment that the rest of the color management chain lives in.
What you then need to do is set the monitor white point to be a visual match to paper white. You should see paper white on screen, the same perceived brightness, the same perceived color. This depends on the ambient light, the print viewing light and your whole environment down to the application interface. So there are no fixed numbers. But for most "average" and "normal" conditions, a white point brightness of 120 cd/m², and a white point color of D65, is a good starting point.
If your display allows it, set the black point the same way: a visual match to max ink.
With this set, you now have a basic screen to print match. What you see on screen will be basically what you get in print.
Now you can run the calibration and profiling. Your monitor profile picks up at this point, remapping all colors to match. The initial calibration will roughly equalize the color balance, but the profile does the real job, at a much higher precision level.
So that's the background. Now for how to do it. With a third-party calibrator that does all adjustments in the video card, it always pays to set the white point prior to calibration. The reason is that the video pipeline works at 8 bit depth, and you need to conserve those bits. If you, for instance, use the calibration to knock down brightness to half, you have already used up one bit, and have only 7 left.
The x-rite software should have a "pre-calibration" function to help with this. You hang the sensor on the screen, and the software measures as you adjust manually. Once you reach 120 cd/m² and D65, stop. Then run the rest of the calibration and profiling normally.
You don't need to concern yourself with the video card specifically. Normally you should just reset it to defaults and leave it. All the necessary corrections are automatically loaded by the x-rite software.
Just to end this, I want to emphasize the importance of the black point, as well as the white point. The black point has a huge impact on the perceived "punch" of the image. This is the main explanation for the myth that "you can never match screen and print". But you can, and a matching black point is a big part of it. Natively, any screen has much too deep blacks.
IIRC, the x-rite software lets you set a contrast range. This is an indirect way to set the black point. If your white point is 120, a black point at 0.4 gives a contrast range of 300:1. This, incidentally, is about the contrast range you get from an excellent inkjet print on high-grade glossy paper.
Oh, one more thing, and this is critical: this is a wide gamut monitor! that means you can only use fully color managed software! No exceptions. Any software you have that doesn't fully support color management - stop using it, uninstall it, hide it away. Don't even open it. You can't use it. Only color managed applications will ever display correctly on that monitor. That's true for all monitors, but a wide gamut one really throws it in your face. The difference is no longer possible to ignore. sRGB will be grossly oversaturated, unless you have color management to remap the colors.
Whew. This became a whole novel. But it's a broad subject.