Let's hit these questions one at a time.
First, how do you split a card into 3 sections? The simple answer is, you don't. You take a single face from a single card. Trying to get both usually ends with one decent split and a second split that has little bits of black glue from the center. Try to get one face from each card. It uses more cards but you get better splits.
Second, what glue do you use? This is a tough one. If you want a really great *Snap* to the card then you should look at Dry Mount Tissue. Derek Dingle published this in The Complete Works of Derek Dingle. At the time it was a WELL kept secret. Today, it is the way to keep things together while maintaining a crisp new card feel. You need a dry mount press or a really good iron and release paper. This is much more of a process but it is quite clean.
Third, how do you line this all up? This is a difficult one. I find that using some sort of reference, like an engineer's square will give you perfect results. It's a perfect 90 degree corner so you place the bottom layer down then you place the glue or adhesive and the last layer goes on top but it starts a little bit out of the 90 degree elbow. Then you slowly slide it into the crotch of the engineer's square.
This keeps the lower split in the elbow nicely and constantly. When the top layer slides into place they are lined up perfectly.
That should keep you going.