Checking the box Enable All "No Chance" and then unchecking it will remove these items from the list, however they return after every pet battle, requiring the option to be toggled again.

Okay this appears to be happening in all zones for all events.

Once again, toggling Enable All Seasonal on and then off fixes this but only until the next list refresh.

Additionally, the tracker is showing non-pets when I only have pets selected for display.

Filters settings page is somewhat counterintuitive and doesn't align with the rest of the settings but to NOT see unobtainables/events you need to toggle them. Think of it as if you are turning on filter that filters these things out instead of enabling tracking for them.
I think we might change it in future.

the tracker is showing non-pets when I only have pets selected for display

You can get Frightened Bush Chicken from Pilgrim's Bounty so it's working as expected (just expand its header, pet should be there).