Currently we just back up the SQL databases on the physical servers. It would be nice to back up an entire image. I understand that the new Veeam Agent for Windows 2.0 (Server edition) which is coming December 2016 will allow a backup of those physical servers to our Veeam repository.
True. Veeam Agent for Windows 2.0 will have full support for Windows servers and residing applications like SQL server.
Can a backup image taken with the Veeam Agent 2.0 Server edition be restored at all, in any way, to a VM on our VMware hosts, or can it only be restored to Azure or a Hyper-V host?
Yes. Export as virtual disks (vmdk, vhd, vhdx), direct restore to Hyper - V and direct restore to Azure are supported.
If it can be restored to a VMware-hosted VM, can this be a "bare metal" type restore which creates the VM in the process, or do we first create the VM with appropriate virtual hardware and only then restore the image?
Bare Metal Recovery to a blank VM will work.
Last question (I hope) -- what Windows Server versions will the Veeam Agent 2.0 support? Can it be run on Windows Server 2003?
Unfortunately, no changes here. Veeam Agent for Windows will support operating systems starting from Windows Server 2008 R2 SP1 and Windows 7 SP1