After months of confusion, I make a post on a forum and hours later find an article that answers (most) of my questions.

The parts I'm most concerned with are in sections 2 and 7. Per the article:

This is where color management comes into play: if applications know the actual gamut of that monitor, they can translate this “0,255,0” sRGB value to another set of RGB values that represent the same color (or fairly close) in a bigger color space: sRGB 0,255,0 (green) -> 144,255,60 AdobeRGB (same sRGB green color)

Since we output discrete RGB numbers to a monitor, usually from 0 to 255 for each channel, and since a monitor accepts a discrete RGB number as input, usually from 0 to 255, then if we modify this one-to-one translation with a calibration curve, we may be introducing “gaps” or “jumps” in that 256 step stair. Such gaps may result in visible jumps between neighbor grey values and even coloration of some grays (red, green or blue tint in them).

So yes, ICC profiles can cause color degradation since they act on 24-bit color most of the time. This also explains why ICC profiles would show up in screen recordings and why you're able to activate color management on a per-app basis since they're just modifying the RGB values to match the standardized color definition with respect to the monitor's gamut/gamma.

TL;DR: ICC profiles are more or less just LUTs that transform RGB values with respect to your monitor's gamut so that the values match the physical scientific definition of each color in the real world. They aren't magic voodoo and are still constrained to the precision of the output signal to your monitor. (but also according to the same article and this Reddit post, some graphics cards have a dithering option that helps)

This doesn't address my questions about monitor settings, but it eliminates the most confusing part for me. If any part of this is wrong, I'd love more information! Until then, I'm marking this as the answer.

After months of confusion, I make a post on a forum and hours later find an article that answers (most) of my questions.

The parts I'm most concerned with are in sections 2 and 7. Per the article:

This is where color management comes into play: if applications know the actual gamut of that monitor, they can translate this “0,255,0” sRGB value to another set of RGB values that represent the same color (or fairly close) in a bigger color space: sRGB 0,255,0 (green) -> 144,255,60 AdobeRGB (same sRGB green color)

Since we output discrete RGB numbers to a monitor, usually from 0 to 255 for each channel, and since a monitor accepts a discrete RGB number as input, usually from 0 to 255, then if we modify this one-to-one translation with a calibration curve, we may be introducing “gaps” or “jumps” in that 256 step stair. Such gaps may result in visible jumps between neighbor grey values and even coloration of some grays (red, green or blue tint in them).

So yes, ICC profiles can cause color degradation since they act on 24-bit color most of the time. This also explains why ICC profiles would show up in screen recordings and why you're able to activate color management on a per-app basis since they're just modifying the RGB values to match the standardized color definition with respect to the monitor's gamut/gamma.

TL;DR: ICC profiles are more or less just LUTs that transform RGB values with respect to your monitor's gamut so that the values match the physical scientific definition of each color in the real world. They aren't magic voodoo and are still constrained to the precision of the output signal to your monitor. (but also according to the same article and this Reddit post, some graphics cards have a dithering option that helps)

This doesn't address my questions about monitor settings, but it eliminates the most confusing part for me. If any part of this is wrong, I'd love more information! Until then, I'm marking this as the answer.

Edit: For those wondering how this applies to DaVinci Resolve or Blackmagic Fusion, it kinda doesn't. See this post.