To be more precise:
1- "I want to collect game scores for each player and keep them in my site member database."
You can do that with the AJAX built-in plugin. You'd have to code the server-side script to set and get datas from the database and return them to your app.
Login screen/authentification is not really advised in JS as eventually, those credentials gould get "easily" intercepted by a malicious person.
But due to the fact that the application can be embedded directly in a web page/website, you could settle for authentification on the website (classic php script/framework) and use session authentification in your ajax communication, preventing the use of the user's credential, but still making sure the user is who he's supposed to be.
2- Once again, with clever skills you could set a turn-based multiplayer game with AJAX calls.
You also have the websocket custom plugin (plugins for C2 section of the forum) that is specifically designed for network communication.
You would still have to code your server.
That's why kbdmaster said it is not multiplayer, because, for now, there's no definitive/reusable example or clear documentation out yet. Is it possible, yes. Is it "easy", not really atm, you need to know what you're doing.
There's a chat plugin available though in the plugins section, and you could maybe consider building upon it, making your own communication protocol for the client/server logic/exchanges. That would make some of the process easier and is reliable.
3- How events work. That's the "programming" you need.
So you could set up a basic IA in events. It is all logic after all.
The complexity of the IA would depend on the complexity you put inside your C2 code, the complexity of the process you set up.
Tom probably could tell more on the subject.