I would have to use three dimensions for my setup, but that isn't my major pain when it comes to the collection it's keeping any inventory location up to date since I have to move the boxes around as the content grows and need to change container. or when new types of parts are added. I try to keep categories together which cause even more moving around...
if I had a software which could help me adjusting and manage that growth.
I have uploaded on GitHub a piece of software that I have developed to manage my LEGO collection. It heavily relies on the Rebrickable API and I would like to thank Nathan and the whole Rebrickable community for having made it possible. My hope is that some of you will find it useful.
You can find it here: https://github.com/aburgueno/lego-collection
And a thread to discuss about it here:
I like the idea. I went with labelling all my boxes with images of the parts, so I can recognize them. Downside is that I don't have a specific order in the boxes (food storage trays, stacked 8 high, 2 rows deep), what makes searching for them time-consuming. If I would number them like you did, that would solve the problem.