There was a panel in the browser, but a few minutes later I turned the screen on and it was gone.
I don't believe so.
TouchPad owners: The TouchPad "How do I...?" thread.
I was wrong. You can. In the top right of your browser there is a Clock/Bookmark icon. Tap it. From there tap the down arrow icon and it will display your active and completed downloads in this current browser's session.