> What is the latest version for the non-Developer version? Where can I download it?
The latest Web AppBuilder for ArcGIS release is embedded within ArcGIS Online. You access and work with it, by logging into an ArcGIS Online organization. Only the Developer Edition is downloadable.
> Where do find out whether it is compatible with other apps? It looks like there are new rules?
I'm not sure I understand your questions. Can you please elaborate, what do you mean by "compatible with other apps" and which "rules" are you referring to?
> Can you tell me if I have the appropriate User types, roles, and privileges to use Web App Builder?