If we program an app on Raspberry Pi, can we duplicate the Raspberry Pi hardware design, manufacture our own boards, and sell them as our product without paying license or royalty for the hardware design?
The hardware design is not "open source". Even if you acquire a license to duplicate the hardware you might not be able to buy any Broadcom SoCs (minimum order quantities in the tens of thousands).
yes you can..... are you really going to believe a stranger on the internet? contact the RPi manufacturer about any licensing questions