1.I believe that in order to use the normal "net use" command, to mount the drive on the server, I would need to be running Server 2012 or greater. Is there any method available to mount the Azure drive on my 2008 server?
In brief, no.
If we want to access the Azure File storage from a computer which is outside of the Azure, the SMB encryption must be enabled for security reasons. And to support SMB encryption, we need to use SMB 3.0. It is first introduced in Windows Server 2012/Windows 8.
If you want to keep a local driver synced with Azure File Storage, then I would suggest you to build a new Windows Server 2012 R2 and migrate your files to the new server.
If you just want to migrate all your files on the local drive to Azure File storage and shutdown the older server, then please use a tool to copy the files.
Are there any Azure tools available to sync the drive? What's the easiest way of going about that?
Microsoft has provided a free software that allows you to access the Azure storage in GUI. To download Azure storage explorer, please click this link.