I really see two key questions here so I will address them separately.
Q1: Do you have to be running a specific Operating System in the Guest VM to create a Replica of that VM in Server 2012 Hyper-V?
A1: Hyper-V Replica is a function of Hyper-V in Windows Server 2012. You can create a replica of any VM regardless of the Guest OS running in the VM.
Q2: For the Hyper-V Replica IP Injection to work when failing over to the replica, are there any requirements for the Guest OS? Just to be sure, I'm referring to the setting in the following screenshot.
A2: For this feature, there are some requirements for the VM configuration and Guest OS.
- You must have the latest integration services installed for the VM.
- You have to use Synthetic network adapters (Won't work for Legacy network adapter).
- At this time, this feature is available for Supported Windows Guest OS's only.
To be clear, Windows Server 2008 R2 is indeed one of the supported Guest OS's so you should have no problem using this feature.
* EDIT - New Information *
I am including a screenshot that shows the difference between a Legacy Network Adapater and a Network Adapter (referred to as Synthetic network adapter in my original answer). When adding hardware, you can choose to add in a Legacy or "normal" adapter. If you look on the left side of the settings pane, you can see how both show up. I hope this helps.