How do you increase the size of it without zapping or getting random events? Or are those the only ways?

You can use a from the Millionaire Center to increase the size by 1 - 3 spaces (random). There's also a Keepsake Box in the Cash Shop, but I always forget how many spaces this one gives ^_^;

To add to what Yiri said, the Multi-Compartment Keepsake Box gives you a minimum of 10 treasure chest spots. :)

Thank guys!