No, it does not harm. DECT phones have an intelligent charger, not allowing you to overcharge the units.
If you are subscribed with a caller ID service it could be that the signal sent by the provider contains the date and time. Your phone will copy these settings.
DTMF (Dual-Tone Multi-Frequency) and the newer version FSK (Frequency-Shift Keying), are methods to send the caller's information (the caller ID).
Digital Enhanced Cordless Telecommunications (DECT) is an European standard for digital wireless Telephones, used in home or office.
You can contact your provider to hide your phone number for the receiver.