Is there anything else in Sql Server 2016 that wants Windows Server
2016 to work well? Or would I be fine with a Windows Server 2012 R2?
There are a slew of new and updated features in Windows Server 2016. Unless any of them are specifically important to you, for example enhancements to storage spaces and CSV storage, I wouldn't specifically call out a need to wait for it.
This will depend, though, on how nice of a system administrator you are. Once Windows Server 2016 comes out, will you build a new system to allow the DBA to migrate?
Maybe you'll want to wait for supportability reasons (granted 2012R2 is still mainstream support)?
There is nothing inside of SQL Server that is really dependent upon Windows 2016 with AlwaysOn. Yes, you have greater flexibility with things like gMSAs, AD-less clusters, etc., but nothing specific inside of Windows that SQL Server hooks into.
Having said all of this, I'd either wait for Windows 2016 to come out or build a 2012R2 cluster for the AlwaysOn Availability Groups and know that we can do a rolling OS upgrade after Windows 2016 comes out. It's double work but you get your cluster now (vs waiting) and can upgrade the same cluster with rolling OS upgrades (so double work).