using same theme for child sites
If you are confident with hacking database tables, you may be able to copy the relevant data into subdomain's database. But there are many hazards involved in doing such.
In general, the design panel is for making occasional tweaks to a page or theme. If you need to go further, creating a new theme by cloning the theme package you started with or shimming it is a cleaner way to work.
If you started with Elemental, there is even a free addon to help you do so - Elemental Cloner -https://www.concrete5.org/marketplace/addons/elemental-cloner...