A translation patch is available.
There's also a phone version (which was released first), but they're pretty different, apparently.
Magic attacks, essentially. The MP-like number shown is the amount of squares you need to uncover for the ability to become available.
In the quest screen there are coloured icons at the bottom of the screen. Apart from enabling different premium skills, they have the follow effects:
These two stages are unlocked by doing things in the mobile version. The only other way to access them is by using a Hex Editor to modify your save file.
See the last part of this post for information:
This post also has info. on disabling the stage unlock timer. You should not disable the timer if it is currently active.