You probably want to use Popup instead of Tooltip, since Tooltip assumes that you're using it in the pre-defined UI-standards way.
I'm not sure why StaysOpen doesn't work, but ShowDuration works as documented in MSDN -- it's the amount of time the Tooltip is displayed WHEN it's displayed. Set it to a small amount (e.g. 500 msec) to see the difference.
The trick in your case is maintaining the "last hovered control" state, but once you have that it should be fairly trivial to change the placement target and the content dynamically (either manually, or via binding) if you're using one Popup, or hiding the last visible Popup if you're using multiple.
There are some gotchas with Popups as far as Window resizing and moving (Popups don't move w/the containers), so you may want to also have that in mind while you're tweaking the behavior. See this link for more details.
HTH.