Yes, you need an Android phone to work with FOWiz Cloud. Just install FOWiz Android App on your phone and forget it. The rest will be taken care by FOWiz Cloud.
All call and text messages are sent from your phone, your operator may charge you as per your tariff.
No, FOWiz do not run in the background and consume your phone battery. Android invokes FOWiz whenever you receive a call or message on your phone.
The communication from FOWiz Cloud to FOWiz Android App is via Google Cloud Messaging System(GCM). Even though GCM offers real-time message transfer, sometimes due to message volume or network problems it takes a while to send the messages.