SQL Server – Login Failed for User ‘domain\localMachineName$’

SSMS uses port 1434 and the UDP protocol. This has to be allowed on the firewall, too.

Quote:

When SQL Server clients request SQL Server resources, the client network library sends a UDP message to the server using port 1434. SQL Server Browser responds with the TCP/IP port or named pipe of the requested instance. The network library on the client application then completes the connection by sending a request to the server using the port or named pipe of the desired instance.

From here.

Yes, it failing is expected. It is documented here.

AlwaysOn Availability Groups and SQL Failover Cluster Instance and Privileges

When installing the Database Engine as a AlwaysOn Availability Groups or SQL Failover Cluster Instance (SQL FCI), LOCAL SYSTEM is provisioned in the Database Engine. The LOCAL SYSTEM login is granted the ALTER ANY AVAILABILITY GROUP permission (for AlwaysOn Availability Groups) and the VIEW SERVER STATE permission (for SQL FCI).

To answer now:

1.I've heard that creating NT AUTHORITY\SYSTEM within SQL server is not safe, If that is the case, Should I change my cluster service to a domain account?, Will this affect anything?

I'm assuming you're talking about the actual cluster service. If that's the case, then no do not change it. You could change it in WS2003 when domain accounts were used but since 2008 this is not possible. If you do this most likely your cluster will break.

2.I added NT AUTHORITY\SYSTEM to Sysadmin role - Is this really required or can I provide only the required permission explicitly to the login

Sysadmins is not required. Per the link/quote above, VIEW SERVER STATE is required.