The E560 handset can be connected to another DECT base, e.g. Swissom. I don't have this, but it should work; I have something similar set up at my parents' house.
Regarding the base of the E560, it should be able to operate the handsets of the C300, possibly with limited functions; but telephoning should be possible. I have registered much older handsets to the E560 base and it works quite well.
Here is the link to the overview of the compatibility between the Gigaset devices:
http://www.gigaset.com/de_de...