Hi!
UVs and UV Sets are two different things, UVs are like coordinates on a map telling your texture where to appear on the surface (of any type, poly, NURBS, subd etc), UV Sets allow you to have different maps that describe different ways for textures to appear on that same object. UV Sets are more or less an advanced feature that not many people have to use (I’m generalising but I work in texturing and hardly ever need to touch them).
When you add new geometry, in the form of an extrusion or bevel or anything that creates new faces, it’s as if a new island rises up from the ocean but our maps don’t know it’s there yet. It exists, but we wouldn’t know how to tell an explorer (or texture) how to get there. As mentioned, inserting edges is ok because it’s like adding a new state or territory to Australia, it doesn’t change how to get there.
To fix problems like the one you showed, you just need to add them into the UV map you already have. You can select a bunch of faces and open the UV Texture Editor and see exactly where they’ve been randomly put (possibly covering the whole UV map), then go into UV component mode and just move/scale/rotate them into a better position matching where the surrounding faces are mapped to. There are much more efficient ways to do it but that should get you started.
Sorry if all that sounded patronizing but sometimes hearing it a different ways helps it makes sense!
EDIT: I suppose UV Sets probably get used a fair bit in games, silly me. I’m coming from the TVC/animation side of things.