Ok, let's address a few things here. First, you want storage inside but want a light. That presents a couple problems. First, whatever you store is going to be affected by the heat of the light. Second, whatever you store will case shadows onto the shoji paper. I don't think that's a good idea. Either do storage or lighting but I wouldn't do both.
Regarding the use of plywood: Are you going to paint or stain? If painting, then ply is probably ok, just sand it good. I prefer mdf or birch veneer and poplar lumber for painted projects.
On the grid, you'll want to use a half lap joint wherever the pieces intersect. That's where you remove half of the material from each piece. A bit time consuming if done by hand but can be done easily with a radial arm or table saw.
As far as the design itself, that's probably going to be a trial and error process for you since you're a newbie and it's something we've all had to go through. Maybe look for plans for something similar but there will be a lot of joinery techniques required for this. It's an ambitious project for a newcomer but it's how you learn. If you make a mistake, learn from it and start over. I've been doing this off and on for 25 years and I still learn things with each project.