The app does make network calls when it starts to get data about servers, players, matches etc but after that it should be stable.
We have added a tone that will play if you have unread notifications, it is not related to searching a game nor does it make any network calls.
How much? While searching its supposed to make a network call every 5 seconds which returns couple of KBs of data.
Progress bar for?
We pushed an update today morning which reduces bandwidth consumption on startup.