Oh dear, that's gonna be a lot to explain, lol.
So basically, MaterialEditor is one of the most powerful tools in KK. In graphics pipelines, a "material" is defined as a series of instructions (shaders) combined with various textures (maps) to determine how light will interact with a mesh.
To put it simply, MaterialEditor allows you to change two primary things: texture maps used by a mesh, and shaders used by a mesh. This can give you insanely powerful control over many aspects of an object's appearance. Beyond just simply clicking through different shaders in the dropdown, you can also tweak numerous settings for the shader itself through various sliders, which further enhance control.
"What the fuck are texture maps?" These provide additional data for the shader to use, which can give you even more granular control over how the mesh will appear. Texture maps will be entirely mesh dependent (with a few exceptions), as they are often built around the UV of that specific mesh.
A UV map is a 2D representation of a 3D mesh, and is one of the most important aspects to any 3D rendering process, so it's very important to understand what they are and how they're used. To put it into perspective, consider the following example of a basic t-shirt:
It starts with the UV Map. The three texture maps used by this shirt are MainTex, NormalMap, and DetailMask. All 3 of these are made specifically for the shirt's UV map. They are combined together by the shader to create the final, rendered appearance of the mesh. There are loads of other maps you can use with specific shaders, like AlphaMasks to cut off parts of clothing, DisplacementMaps, HighColorMaps, OutlineTex, the list goes on and on and on.
What do all these maps do? I can't describe all of them as it would be far too exhaustive, but the most basic maps are the three from the above example. MainTex would be a diffuse/albedo map under typical industry terminology. This contains color data only. DetailMask would be pretty similar to an AmbientOcclusion map. This creates faux-shadows on the mesh. NormalMap is sort of like a BumpMap, it adds subtle depth to features that would otherwise look completely flat.
Other maps, like AlphaMasks, MatCaps, LineTex, etc are used for specific shaders to further alter details. There are so damn many possible options I can't list all of them, but if you're ever unsure what a particular texture map is doing, export it and look at it. If it still doesn't make sense, delete all the pixels and load in a completely empty version of it to see how the mesh changes.
The confusing part is trying to wrap your head around editing a 2D texture for some effect you want, when you don't really know what it'll look like until you throw it all into the MaterialEditor. This is where Substance Painter is really handy, because you can export OBJs and UVs of any mesh and throw it into Substance Painter. This makes visualizing the apperance of the texture maps a lot easier to understand, because it literally shows you how it's gonna look wrapped around the mesh right there.
Now we can finally segue into your issue with copying shaders. Often times shader configurations you'll wanna copy contain overrides on their texture maps already. You can tell if a shader for a given mesh contains map overrides by simply looking at the MaterialEditor:
In this example, all three of these texture maps have been altered, as indicated by the slightly darker background in their respective fields. This means if you hit "Copy Edits", these texture maps will tag along with that copy. This is why you run into issues when copying a shader to a different object. The shader contains custom maps, and when those same maps are applied to a different mesh, they don't line up right because that mesh is using a completely different UV map.
For the most part, this is not an issue as a majority of the time, you don't need the maps, you just want the basic shading appearance from the rimlighting, outline settings, reflect blend modes, etc which will largely be determined from the slider settings used for that shader. There are some exceptions to this. MatCaps and Hair Gloss for example are not UV-dependent so should always be copied. But sometimes a piece of clothing only looks that good because the creator has some insane texture map they whipped up. That is where you'll need to attempt to rebuild what they did.
To copy a shader without moving those custom maps along with it, you can just click the "Reset" button on each of the overwrriten maps. Once this is done, you'll have to save the scene (if doing so in Studio) and reload it, or simply change outfits on the character.
Upon reloading the scene or switching back to the outfit, you'll see those fields no longer have a darker background, indicating they have not been "edited", so they will not be copied. You can then safely paste edits to whatever you want. The overall appearance of the mesh should be almost identical to the thing you copied it from. If you find particular details to be lacking, you can export the various custom maps from the original to see what might be different about those maps, and compare them to the ones on the thing you're messing with.
I have a "template character" with a bunch of different shaders for various kinds of clothing and hair objects loaded on different outfits. This template character was created by simply using Studio to load in characters with shaders I liked, resetting textures on the shaders I wanted to copy, then pasting them onto the "template character". Once I had all the shaders copied with no texture overrides, I then exported that template character and popped it into Maker to perform a permanent save.
Any time I want to copy some nicer looking shaders to anything, be it clothes, hairs, or a character's body, I just pop this template character into the scene and I can very easily copy/paste shaders over to whatever I want.
As for saving scene presets and post-processing presets, this should be a feature already available to you as I'm pretty sure these are default plugins with the HF Patch. The two plugins are "Scene Effects Presets" and "Save_PostProcessingEffects". If you open the F1 menu in Studio and search for either of these, they should pop up and allow you to set keybinds for them. If you don't have these, they can be found very easily in the KK Discord or from googling.
And lastly, as for overlays, think of these as separate from MaterialEditor. Underlays overwrite MainTex of the body/face and will be influenced by color data from the Skin Color of the character, assuming it is even slightly transparent. These don't function as copyable edits within MaterialEditor. The actual Overlay slot goes on top of the MainTex.
I tend to rely on basic skin coloring to give the MainTex a specific hue, and use detailed overlays containing features I want (veins, skin detail lines, blemishes, blushes, sweat, hair, writing, etc) on the Overlay part. Basically, think of the Underlay slot as "color tintable", while the Overlay slot is not.
Some overlay packs (like what you've linked) were built with specific texture maps in mind, so to get the most out of those overlays, you'd need to import the various texture maps it comes with onto your character, tweak the shader settings per their instructions, and then slot in the necessary overlay/underlays.
From here you can simply save this character and use it as a foundation to copy shader settings to any other character you want, the only step you'll have to do is slot in the Overlays/Underlays. In this situation, you WOULDN'T want the source shader to have reset texture maps because you need them.
Texture overrides go both ways, too! Suppose you have two characters, and both have custom maps for DetailMask and NormalMap. You need these two maps from your source character to go over to the destination character. In this case, you'd just have to reset the textures on the destination instead of the source, and voila.
