What isn't mentioned anywhere here is whether the Start Editing option is being used or not.
I agree with Tom that I can't reproduce the first issue. If that is consistent, further troubleshooting with technical support would be an appropriate avenue.
As for after I use Calculate Field, the 'Save' button in the 'Edit' tab is not activated:
If I calculate a field without enabling Start Editing, this is what I see:
The Save and Discard buttons are disabled because the calculation was not done in an "edit session."
If I run a calculation with Start Editing enabled, when the calculation completes, here is what I see:
Save and Discard become active and I also see the GP pane update with a banner telling me that I have pending edits.
So whether Start Editing is enabled or not is crucial to this conversation. Thomas Colson you said you simulated a crash (like what? just Pro down without saving anything?) and the Calculate Field edits persisted. Did that occur even if you Start Editing, Calculate Field, do not save edits, close Pro and choose NO to not save edits, then open the feature class again?
I don't see that edit persist: