Can you load a widget instance from a template?

Yes, you can do something like this:

$widget = Mage::getModel('widget/widget_instance')->load($widgetId);
$widgetBlock = $this
                   ->getLayout()
                   ->createBlock($widget->getType(),
                                 $widget->getTitle(),
                                 $widget->getWidgetParameters());
echo $widgetBlock->toHtml();