Yes and yes. I don't recall setting those options, are they the default values? If I turn this off, is there a way to find broken links manually?
They are default settings, and I'm not aware of a manual option. When I need to fix broken links (which is very rarely, because I do all of my media file management using MC's library tools), I just backup the library, turn off auto-import in the background, turn on "fix broken links," and run a manual auto-import. It will tell you in the summary if it "fixed" any links during that import, and list them. You can confirm that it only fixed the links you wanted fixed.
the recent albums view seems to be correct, the recently imported view is a random list of stuff that I imported ages ago. If I look the "date imported" tag though then the recently imported view doesn't correspond to the most recently imported stuff so I don't know what is going on there.
When MC "finds" a file that was previously in the library it reimports it and tries to attach it to the old library entry with the old metadata. So the date imported will be the original date imported (sometimes), if MC correctly identifies the linkage. If MC doesn't match it up to the old file, it will import it "fresh" and assign a new date imported. You're probably seeing the former case, and the mismatch between date imported and the recently imported view is confirmation that you're experiencing a "fix broken links" issue.
If things were being removed by that fix broken links setting but I have auto import on then shouldn't it have automatically reimported them or does it drop entries and then ignore them in future?
My experience is that they usually get reimported eventually, but not necessarily automatically for whatever reason (sometimes they would get picked up on the next manual import I ran). Ultimately I found the troubleshooting process so onerous that I just disabled fix broken links (it becomes especially hard to diagnose in a client/server scenario). Use the rename/move/copy tool to do file management, I very rarely have any broken links to begin with.
I've had no issues with any of this since disabling "fix broken links."