If you're gonna be doing it that small, why not build it in a PC shell. like the ones they have on ebay for 600.00. im sure you could get an older PC for free if u looked hard enough. gut it and built your box. just a suggestion
yes most likely is there a way to build a small carbon filter to harness the smell or any way i could keep the smell in the container? and when you say grow it for a week you mean germinate the seed wait til it sprouts and that be the begining of the week?