I have completed the "Build an App" tutorial here: https://www.polymer-project.org/2.0/start/toolbox/set-up and have noticed that the paper-checkbox element is not behaving well - it will intermittently freeze at various stages of the re-render. For example, when it is in the unchecked state, and I tap it, it will freeze with the grey circle surround, but without actually displaying the tick in the checkbox. It will also freeze in other states as well. I think the problem only occurs for screen gestures. I.e - it will fail on a mobile device for screen taps, and also on my Win10 laptop when I tap the (touch-sensitive) screen, but when I use the touchPAD on my laptop, it works reliably.
I am a raw beginner with Polymer. (and everything ;^)
This seems to be related to a known issue (PolymerElements/paper-checkbox Issue #156).
I can reproduce the buggy ripple animation on your site in Chrome 66 on macOS High Sierra when emulating mobile mode. It seems to intermittently occur after switching views (e.g., to View Three and then to New View) and then clicking the checkbox label (clicking the checkbox itself does not exhibit the problem).
Part of the problem is the checkbox's ripple has a "focus" state, where it shows the middle of the animation (the "frozen grey circle" you mention) when the checkbox gains focus. I'm not sure that's by design.
I recommend voting on the GitHub issue, or creating a new issue if you think it's a different problem.