What is the "Pushed tasks" folder and why are those items read-only?
It's a folder in the sidebar (and a real "Pushed ✓" list on your device) where every handed-off task is filed, grouped by the list it came from. It appears whenever you have a task app connected, even when empty. Items in it are view-and-delete only because they've already been handed off — there's nothing more to push. Ticking one done here doesn't change the copy in your task app.