A "Sync progress" box with a log and a timer appeared during a push — what is it?

That's the push-progress window: a live log of each step, a running timer, and a progress bar so you can see how the hand-off is going. While a manual push is running, open it with the little arrow (›) that appears on the push pill in the bottom left — it records the whole push from the start, so you get the full log even if you open it midway. It's only available for pushes you trigger yourself, not automatic background pushes, and otherwise you won't see it. Close it with the × in its corner.