You can also hold down the Shift button and hover over it and it'll immediately show.

Something that would be really useful would be a permanent icon to drag and drop that always refers to last downloaded file. e.g. an icon in the task bar that I can drag into an application or a finder window that will copy that last downloaded file there. Any idea if that is possible?

defaults write com.apple.Preview NSToolbarTitleViewRolloverDelay -float 0 for Preview, too. It would be amazing if there was a global setting for all apps that use NSToolbar... :(