Does Kongregate allow us to use store persistent game data (not statistics) in its server, or are devs expected to have an external server to store those on?
Yes, you can store persistent data via the Shared Content api. Unfortunately, that API is targeted towards actually sharing content that a user created rather than storing information about a user like a save game or other general user properties. (That doesn’t mean it can’t be done, just that the solution won’t be super nice)