Long story short, I was copying my projects to a new system and there was a bit of a failure with some files in the Library folder.
My understanding is that these are files Unity created, since, well, I didn’t. Is there anything I should be doing to make sure these files are properly reconstructed by Unity? Is there anything I should be doing to make sure Unity recognizes that these files are missing?
Just what is this Library folder for anyway? What are these files in it?
You do not copy the library folder around. It is the transiently-built parts of your program (eg, what happens at “import”) and the reason you can press PLAY and Unity instantly begins playing.
Rather than copy stuff around and never be sure if you caught all your changes, consider using proper industrial-grade source control in order to guard and protect your hard-earned work.
Personally I use git (completely outside of Unity) because it is free and there are tons of tutorials out there to help you set it up as well as free places to host your repo (BitBucket, Github, Gitlab, etc.).
As far as configuring Unity to play nice with git, keep this in mind:
https://discussions.unity.com/t/736093/3
Here’s how I use git in one of my games, Jetpack Kurt:
https://discussions.unity.com/t/807568/3
Using fine-grained source control as you work to refine your engineering:
https://discussions.unity.com/t/826718/2
Share/Sharing source code between projects:
https://discussions.unity.com/t/719810/2
Setting up the right .gitignore file:
https://discussions.unity.com/t/834885/5
Generally setting Unity up (includes above .gitignore concepts):
https://thoughtbot.com/blog/how-to-git-with-unity
It is only simple economics that you must expend as much effort into backing it up as you feel the work is worth in the first place.