DIY Portable Grow Box 7x3x6 Concept

I think your  setup looks great as of now.  just like ice_cream_man said be very careful when it comes to your wiring man. my buddy set up his grow room and almost burned down his  house.  lucky he was in the room when it caught fire and put it out hastily. needless to say the dumbass could have had  asked our best buddy who is a certified electrician to fix the wiring for him in the first place. next day hes was there rewiring the panels  and basement.  you don't want to end up dead because of something you could have prevented. before you start construction just  go on graigslist and put a listing up for an electrician to come check out your breaker box and tell him what you need and have him set it up for you before anything. prolly only cost you like 60$

 

Looks like youre planning this out properly.
 
One piece of advice I will lend as you havent actually started building yet would be to invert your seedling chamber to the bottom. The seedling chamber will run on fluros so you wont have too much heat to deal with.
 
This places your 2 high power lights at the very top of the box. This will help with ventilation quite a bit, you could even run the exhaust for both chambers through the middle section so you only need to run a single fan, just need to duct it so that there is no light creeping to the flower chamber.
 
your standard 12-2 wire can handle 20a no problem, so opting to just swap in a larger breaker should be safe.
If this box is running on a relatively isolated circuit, 20a should be more than enough to run the whole setup. If you get into heaters or ACs, or excessivly high powered fans, you may need to opt for a 30a circuit, but this will require running larger 10-2 wire minimum from the breakerbox.
If your planning on putting this cabinet anywhere near your breaker box, or in the same room, just run it on its own dedicated circuit. It may require running a bright orange wire along the corner of the room to it (unless you want to tear up your walls to install the wire properly) but thats no big deal if its in a utility room or garage or something...
But I think 20a will do just fine for you. My grow cabinet is in my bedroom which only has a 20a circuit and I run room lights, grow lights, fans, a space heater, phone chargers, computer, xbox, tv, modem/router, and stuff like that and still dont have to worry about tripping a breaker when I plug in the vacuum to do some cleaning. So I think you will be fine..
 
Looks like you will have a sweet setup for yourself either way.