I was installing a couple of mods, and got a message from NMM saying something about the ini edits depreciating and causing long load times. It recommended deleting the old file and redoing the edits using an updated guide, but I didn't create a backup, since I didn't expect to have to change these. Would verifying game integrity via Steam get the old INI back, or do I need to do something else?

Usually, you delete (or rename) the ini file and it will be re-created.

Is it one of those "Start the game through Steam to get the launcher, then close the launcher instead of hitting Play" things?

Probably best to just start the game through the launcher and go as far as the title screen. Probably have to redo any changes to the gameplay settings or control mapping you've done though.

Ok. I'm just afraid that starting the game without the edits will screw up my save. If I'm just going to the title screen and not loading the save, though, I should be fine.

Unless you made changes to the uGridsToLoad setting (which is inadvisable, and you'd know if you did), inis should have nothing to do with your saves.

Not in and of itself, but my understanding is that the INI changes allow mods to load. Load a game without the INI edits that allow mods to load is like trying to load a save that was made with a mod active after removing that mod. At least, that's what I'm assuming. I've been doing PC gaming for a while, but it's mostly just been bumbling my way through, so I'm not actually sure how a lot of things actually work.