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)
<img src="http://new.tinygrab.com/a160021eefafcbc8841cce3c4dd8f0082d42db8341.png" border="0" />
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