Let me tackle the easier stuff first:
1. I love the idea of hi-lighting the arrow button, consider it done!
2. I completely understand the issue with the buttons. Given that someone might go through it all the way, then want to redo parts of it, do you think the following would be ok?
a. Buttons remain hidden initially
b. Appear after the last character (whether it's kanji or vocab quizzes) is finished at least once, and remain on the screen after that (even if they go back and redo some or all of the characters).
3. Jumping to the grading thing, I know exactly why it's happening, but I have less of an idea as to how to approach it the most effectively.
The reason it is doing that came from when I originally designed the question modes where the characters disappear. The idea is that say you draw a perfect stroke, but it's 5% (of the box's width) too far to the left: it shouldn't be marked incorrectly. So the system has an offset limit - if it is within x% of the starting point of the stroke, move the stroke so it lines up with the beginning.
The biggest problem with kana are the long strokes - there is a good chance that you'll start off, but as long as you can see the line, you'll adjust yourself and fix it early on. However, the beginning of the stroke is off, so it moves the entire thing.
I think for now, I'll just turn off the offset system if the stroke is still visible.
I'll get #1 and #3 done soon, and will consider #2 after hearing back from you.