Advice on building small smoke house/box

You are going to want to use this then for just "cold smoking"?
Unless you plan on doing straight cold smoking with this, your temps MUST be above your stated 120 degrees maximum.

I used the 3/4" plywood to build my upright smoker, I use a propane burner off of an old grill. I initially wanted to use the hot plate too but, I was only able to get the temps to about 140 max with a 1,000 watt hot plate. That does not include the water pan or any of the meat, which is why I chose to go with the propane burner. If you go with the propane I would suggest looking into the afterburners, check this site out http://www.gassmoker.com/index.htm My box is 48" tall by 26" wide by 24" deep. I built it to be able to smoke my sausages and also do large cuts of meat like pork butts, ribs, and briskets etc...
I did not insulate my box, I live in MN and have used it several times this winter. I dont have it vented out side so I had to smoke in the garage with the garage door half open for the smoke to escape. I am still able to get temps up to 350 degrees in the winter which is more than enough for what I do. If you wanted to solely use the hot plate I would not go any taller than 36". The taller you go the harder it is to heat all the space so, if you are just going to be using it for several cuts of meat and not doing sausage where you have to hang it you could actually go a little shorter if you wanted too. Hopefully this will get you started, good luck!

which burner did you use with your wooden box cooker

you might look at garage sales or on ebay etc. for a big chief smoker. close to your dimensions and the workd is already done. it is great for what it's designed for, which should be exactly whatr you will be asking from it.