The Gigaset N670 IP PRO is compatible with the full range of Gigaset professional handsets, but the CL660HX is not explicitly mentioned as a compatible handset for this base station. The CL660HX is a DECT handset designed to connect to DECT-enabled routers and base stations, but it is not part of the Gigaset PRO series, which is intended for the N670 IP PRO.
Therefore, it is very likely that the CL660HX is not officially compatible with the N670 IP PRO base station. For secure compatibility, handsets from the Gigaset PRO series should be used, such as S650H, R650H, SL750H or Maxwell C, which are explicitly supported with the N670 IP PRO.
Analogue telephony is not possible with this device. It is a DECT base station that is designed exclusively for connection to a VoIP telephone system (VoIP PBX) and supports IP telephony. The device connects DECT handsets directly to an IP-based telephone system via Ethernet, not via analogue connections.
No, not per default. Your DECT handset must send out unique identification number to be registered. It is not that simple as with normal base stations.