As You can see from Software log, phone's modem part is damaged, which is most likely:
Most likely this is the reason for problems with Network. Damaged modem part can be Software (modem firmware or EFS is damaged, or You have written unsuitable firmware) or HW problem.
You should re-flash the device with different firmwares one by one and check, whether modem is correct (not Unknown).
"Repair Network" operation requires phone to be rooted:
Quote:
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Can't get root privileges.
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