Moving and styling menu

It should be possible "move" the menu used in the default templates ("month","week"), and display it elsewhere programmatically. It should be styleable without affecting other site menus.

Currently the menu appears to be using the Navigation menu, and cannot be changed without losing major features: It will only display if the default display has a menu item entry setting of "default menu tab", which does not allow its location to be changed to a custom menu. If it is changed to any other value, the menu will not display and/or the base path feature (mysite.com/my-path) is disabled.

For example, I need the menu to be below the date heading in the pager ("Monday, December 1, 2014") without losing the base path feature of default menu. I also need to style the menu and its tabs, adding icons and display a description for the menu itself. This should be doable in a template file, ideally.