A lot of those questions are answered in their FAQ. The Duple application on your 'server' will simply expose a directory on that system (or a remote SFTP serv

A lot of those questions are answered in their FAQ. The Duple application on your 'server' will simply expose a directory on that system (or a remote SFTP server) to other Duple applications you connect to it. All data will be on the machine you maintain. So no, they are not storing your data for you. To enable access to your device from outside your local network, you have to either give the Duple running machine a public ip and open the port, or do port-forwarding if you are behind a NAT firewall (which is the most common situation for home cloud stuff)