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I always get this message while using STAMINA. Yes, GMAIL (the native email system on the android) has been added to the "apps active in standby" list - makes no dif.
All other apps on list function properly, just not GMAIL.
Any ideas?
BTW, like a few others, I also have "Network Unavailable" issues, and WIFI connectivity "weak/dropped" issues...only with the Sony Xperia Z, not with any others (iPhone,HTC, Samsung...all fine)
Model C6606
AV: 4.1.2
BBV: 9x15A
KV: 3.4.0
BN: 10.1.1B.0.166
Adding apps that require to be syncing or internet access to that list will defeat the purpose of having stamina mode, think about it.
Don't forget to give Kudos
As for your other issues, perhaps repairing or update your phone to the latest available firmware might solve the issues.
Update Service (SUS)
http://www.sonymobile.com/gb/tools/update-service/
http://www-support-downloads.sonymobile.com/Software%20Downloads/Update_Service_Setup-2.11.12.5.exe
Bridge (for Mac)
Alternatives on How to backup Xperias
http://talk.sonymobile.com/thread/36355
Don't forget the Kudos