Sometimes a mod will crash but recover itself, while an exception is printed to the log. Could you add a feature to print the mixin class there?

I don't think this is really feasible, as I can't hook every place where printStackTrace or similar JVM functions are called; interoperability would have to be written for every single mod that prints exceptions to the log.

When does this actually trigger? I'm having trouble actually getting a crash report, as sometimes in the log Minecraft says it crashed but no crash report was generated.

It triggers whenever a crash report is generated; when you see the ---- Minecraft Crash Report ---- heading. I think this is usually shown in the launcher's game log (I suggest using PolyMC/MultiMC/ATLauncher, the vanilla launcher doesn't always show the actual crash log) as well as a file in the crash-reports folder.

Can minecraft crash without generating a crash report?

Yes, though it's uncommon since most Minecraft code paths catch exceptions and generate a crash report.