SQL Server 2012 with account NT Service\MSSQLSERVER access is denied in domain
Those three problems are all a result of the account running the SQL Service not being a domain account, and they will all be corrected by changing SQL to run under a domain account. Specifically:
A - an SPN is a Kerberos security feature that requires a domain account, and doesn't work with local accounts
B - In order to read from active directory, the service needs a domain account's credentials
C - Local accounts aren't recognized by remote computers, so they deny the connection attempt.
Here's a walk-through on how to change the service account:
Old question but doesn't seem to have a proper answer. NT Service\MSSQLSERVER being a local virtual account, it accesses the network as the computer account. And as long as the computer account has access to shares and filesystems you should be able to for example backup to UNC paths on the network. See Configure Windows Service Accounts and Permissions.