I flashed Philz CWM recovery, wiped data,cache,dalvik, then I flashed newest CM11 (12.07.2015) and still "Emergency calls only".
Still in Settings->About phone->Status Imei is 0049... (but Serial number changed from 000... to a random number)
And If I understand correctly efs and imei problems occur mostly during rom change - but my problems started during normal use of my S3 (I was downloading GTA China Town from Google Play store when my phone stucks and didn't reboot correctly, then info about corrupted efs etc...)
Maybe my problems are somehow related to error which I had with ADB?:
So I flashed CWM recovery, connect with ADB and used commands:
mount -w -t ext4 /dev/block/mmcblk0p3 (at this command I had error: can't find /dev/block/mmcblk0p3 in /etc/fstab)