Yes if you use local storage it should just create the key data on the mobile device in the same way. Once you have it working with the preview (run layout) on the PC you will likely see it working on the mobile app unless something went wrong along the way with the third party plugins, I guess that would be a separate issue. Ensure it is working on preview browser first, if your data is refreshing every time then there's a problem with your events.
In terms of what you're describing for the save data. Couldn't you just save one variable that unlocks stages of your game based on what the variable is? i.e. if the variable is 3 then levels 1- 3 are unlocked.