Simple test for gradiant fill on a Panel with a Theme override and all it took was a one line addition to the CSS of the panel control.
For the rounded corners, I'm sorry but my memory is fading. In order for you to be able to simply switch out (override) images for the borders, the control has to be implemented in Html using the niner approach (Top left, Top, Top right, Bottom left, Bottom, Bottom right, Left, Right and Center). This is currently not implemented for the Panel control, but I know it's implemented for Button and Form.
This means that for rounded corners of a Panel, you would need more customizations, both for XSLT and CSS, see here. It might be possible for you to use a non-dialog form in place of the panel and customize your form to show round corner.
If none of that helps, I think be are back to the step of finding a way for you to invoke the rounded corner jQuery code when the Flash control causes the refresh. If at all possible, please email the sample application to support so we can at least take a quick look.