1) Uncheck this box in your game's settings - then you can call up the login window with the action "Prompt Login" (I just updated the plugin with that action, as well as a condition for when they successfully login after you prompt, and an expression on if they're logged in or not)
2) They're automatically synced. They're also synced if they haven't joined Clay.io and join later.
Leaderboards do not work offline - I suppose it's possible, but I can't say it will be implemented any time soon (so many other things to do, and we're a very small team). For it to work, you would have to include socket.io and our API (downloaded) in your project files, and we'd store the scores in localstorage until a connection is found.
It should be using your Clay.io login (perhaps it uses the same gravatar image which is why you thought it was your Scirra login?) - you can test it by opening an incognito window and trying again (if you're using Chrome)
Yep, the API will work on Chrome's Web Store