Here's my version, 1-1 to 3-3:
http://dehacked.2y.net/microstorage.php/info/1000307159/Puzznic%20%28U%29.fcm
I don't intend to continue this. I just did it for fun and as a demonstration of what is possible in a TAS. I know of several improvements, including the obvious improvement between levels which I somehow didn't notice while making this.
It turns out that there is a faster way to move the cursor in this game, which I stumbled upon while working on 3-3. If you press any directional buttons that you didn't press in the previous frame, the cursor moves in the direction you're pressing, including any directions you pressed in the previous frame. (I'm not sure how to word that in a non-confusing manner.) Thus, if you alternate up and down while holding right, you'll move to the right every frame instead of every other frame. The same applies for any direction. In some cases, you may want to start by pressing the direction alone for one frame for parity reasons.
Good luck with this run. Puzzle games can be very difficult to TAS, simply because they're designed to be complicated to figure out normally.