That sounds like a plan....although what type of stain do I want? And what tint would be ideal for a very light coloured birch?
For the wax, I figure beeswax is fine, or any furniture wax.
When it comes to sanding and painting MDF, I have tons of experience. I'll use 1000 grit on the cut edges of MDF to make them real smooth. The true word, in french, would be lisse.
I have worked with pine and cheaper woods before, and in my experience, they don't require nearly as fine a grit or strong pressure to do the job. I was thinking of first sanding with 150 grit, then moving on to 300 before I stain, and then maybe a 600 before waxing. 1000 grit seems like it may make the surface too smooth.
I forgot to mention that I will be assembling the box using 45' mitre cuts all around to make sure there are no edges. I will be fastening the box with glue and clamps, and possibly some brad nails. I want this thing to look as clean as possible, hence why I'll be opting for a buffable wax.