You are right only one of the routers should have DHCP enabled.
The ISP won't know the router password especially as it appears to been changed from the default.
There will be a reset button, or small hole, on the Gigaset which will reset it to the default settings. If I read your post correctly this is the router actually giving you the connection to the whole remote set up? So if you reset you will loose connection.
Also the routers appear to be on different ranges - third number - ip address one is 5 the other is 0 - need to be the same.