The question is very broad and probably beyond the scope of an answer here. Entire books have been written regarding points 1-5 of your question. It seems that all of that is really just a long preamble to what it seems you may be really asking:
Why sRGB images viewed via the Chrome browser do not shift colors when I change the color space on my system to Adobe RGB but sRGB images viewed via Lightroom do shift colors when I change the color space on my system to AdobeRGB?
I cannot see any changes at all when switching between AdobeRGB and sRGB as the default color space. I don't understand why. Shouldn't images open in Google Chrome change as I change the color space in Windows Color Management?
Chrome is a color managed browser. That means that if the sRGB images in question include the proper color profile information in their metadata they will be correctly displayed regardless of whether your color space is set to Adobe RGB or sRGB.
When I open Lightroom, images that I have edited to be exactly as I want them on an sRGB 100% coverage monitor are way too blue on my new AdobeRGB 100% coverage monitor.
That's probably because you have either told Lr to use Adobe RGB regardless of the color space indicated in the metadata of the photo or because the correct color profile information was not included in/has been stripped from the metadata of the sRGB images and in the absence of a specified color space Lr is using Adobe RGB as the default.
From comments by the OP:
So what you are saying is that if I set Windows Color Management to use AdobeRGB, and the monitor itself to AdobeRGB, colors should be correctly displayed in color managed applications, and too saturated in non-color managed applications? So in Chrome an image with an sRGB profile should look as dull as it does on an sRGB monitor?
sRGB is a smaller gamut than Adobe RGB (there should be a space between "Adobe" and "RGB" - adobe.com/digitalimag/adobergb.html). But they both use the same number of steps between "0" and "100%". So the number that equates to "88% green" in sRGB will be a different color than the same number that equates to "88% green" in Adobe RGB. What color managed applications do is translate the numbers from images encoded in one color space to the numbers for that same color in the other color space. If the color is outside the other color space, though, it can't be displayed. How out of gamut colors are translated depends on how the application has been instructed to deal with out of gamut colors, either via the default settings for the application or via a user selected setting. Relative colorimetric will only shift the out of gamut colors to the nearest color that is within the smaller color space. Perceptual will shift all of the colors to avoid stacking all of the out of gamut colors at the "edge" of the color space which can lead to some weird artifacts. An sRGB image only contains information about the colors within the sRGB color space.
And an image tagged with AdobeRGB should be correctly displayed on my monitor, while be displayed too dull on "regular" sRGB-monitors?
If the Adobe RGB image contains colors that are out of gamut for sRGB then they will be shifted to fit in the sRGB color space. How they are shifted depends on whether "colorimetric" or "perceptual" (or any other choices the application may give you that are variations of those two basic approaches) rendering has been selected by the application or the user.
Also, the images in Lightroom are RAW images. Might the problem be that since Lightroom edits RAW images in ProPhoto RGB, an image that looked good on an sRGB monitor looks overly saturated on my new AdobeRGB-monitor, simply because my old monitor couldn't reproduce the colors really saved in the RAW image?
Raw images have no color space. They are monochromatic luminance values. When you "view" a raw image on your monitor you're looking at one particular translation of that raw data. For more please see: RAW files store 3 colors per pixel, or only one? and While shooting in RAW, do you have to post-process it to make the picture look good?
Apart from color space, Lightroom also applies its own default rendering when you open a raw file with it (you may have changed the default profile to one you created). But the data in the raw file is neither changed by the settings saved alongside the raw data nor is all of the data in the raw file displayed at once on any monitor, regardless of the color space. There is no correct "one" way to display a raw file the way there is only one correct way to display a jpeg image.
If you've edited and "saved" a raw file in Lr, the actual luminance values for each pixel haven't been changed. You've just saved a set of instructions on how to display the raw file the next time it is opened by Lr. Things such as WB, contrast, brightness, sharpening, etc. are applied when LR opens the raw file again based on the settings you saved.
"...simply because my old monitor couldn't reproduce the colors really saved in the RAW image?" No monitor can display all of the information saved in a typical raw file. Even the widest gamut color spaces are limited by the capability of the display medium (monitor/printer). It was the same with film. No printing paper was capable of the same dynamic range as the negatives. That's why what Ansel Adams did with the zone system was considered such genius: he figured out how to use lower contrast paper to show a higher contrast scene in a way that looked natural to the eye.
So in order to make sure my images are saved correctly, they should be edited in AdobeRGB (RAW) instead of ProPhoto RGB. However at the expense that in the future when the monitors get better and covers more of ProPhoto RGB, my old RAW images will not use the full color space.
As we have said above, raw images are linear monochrome luminance values. They have no color space. When you export an image it is no longer a raw file.¹ It is either a jpeg or a tiff or a png, etc. For those you have to select a color space and live with it. That's why it is important to save the raw data as well. The raw data will allow you to leverage future improvements in display technology. But you'll have to redo your editing of the raw data and export it in the new color space to take advantage of it.
Remember, even when you are editing in ProPhoto RGB what you are seeing on your screen is being translated to sRGB or Adobe RGB or whatever your system's color profile is. The ProPhoto RGB is the internal color space, but it isn't being displayed unless you have a system and monitor capable of displaying 100% of the ProPhoto RGB gamut!
¹ This includes the rendering you see on your screen when working with a raw file. What you see is not "THE raw file." It is an interpretation of the raw data based on the current instructions (color temperature, white balance, contrast/light curves, brightness, sharpening, noise reduction, etc.) on how that data should be converted to a viewable image in a particular color space.
"So this means that Lightroom will convert ProPhoto RGB to Adobe RGB on the fly for the monitor? So this means that if I open the same image on a ProPhoto RGB monitor, the image will look a lot more saturated?"
So I should really choose Adobe RGB in Windows Color Management as long as I don't use calibration hardware?
If you don't use calibration hardware you're just chasing a wild goose. Do whatever makes the photos look the way you want on your system. But there's no way to predict how they'll look on any other system, even a properly calibrated and profiled system.
How much will calibration do on a $1200 factory calibrated 5K display? Will it be noticeable to the naked eye?
Probably. Even if not immediately out of the box, it will eventually. Calibration isn't just about what it looks like when it leaves the factory. It's about how every monitor changes over time as it is used. It's about matching the output of the monitor to the viewing conditions (ambient light). There are a lot of factors that can't be accurately accounted for at the factory.