First playable version
After I solved the whole how-to-store-those-damn-pieces thing my progress got much faster 
I ended up implementing commands to move/rotate the pieces like regular tetris to help debugging before making the AI (hardest part by far). Those are via USB to the Arduino so I don’t have to mess with hardware buttons again. You also have the physical button to change the next piece (I do it a couple times on the video below).
Tetris is somewhat playable but with some of the twists of Sirtet (the name of my game!): you gain negative score when a line is filled and positive score when the top of the board gets filled. As of now it has no restart option (on the final version you’ll just gain points from making the AI fail and speed will increase).
Here’s a video (forgive me for the shaky camera, poor lighting and the damn LCD glitches):