Thank you both for your ideas! Following them, and tinkering for a while (and going mad sporadically), I eventually did as much a clean install I could:
1. Created a clean new project in Unity 5.6.6, then installed the newest Playmaker 1.9.0p2 from the Asset Store + the same for Ecosystem
2. Then, one by one, manually, moved my other addons (and their .metas) from the old project to the clean project, checking after each move that everything was still working expected by relaunching Unity after each move I made
3. Then, brought in my actual project files (scenes, prefabs, etc.) and their metas in the same way
4. Then, installed any custom actions from the Ecosystem that were now missing
5. Then, deleted each old addon I had one by one and installed the newest versions from the Asset Store (also one by one, relaunching Unity after each change)
6. Fixed all the little things that came up with the new versions of the addons
End result: everything is now working as intended and there's no NullRefenceException errors anymore.
Needless to say, I backed up the project after each move I made, so it took a sweet while to do all this, but at least I didn't accidentally destroy anything forever and can now continue the actual development. (Well, I guess I'm gonna try to update to Unity 2017 and 2018 before that, but that's gonna be another story.)
What was nice about this spring cleaning that I also got rid of old legacy scrap from Playmakers before 1.8.1, and custom actions and packages I didn't actually use anywhere.