Well, thats hard for me to say for sure. I moved away from plastic containers for the roots almost immediately when I started doing HPA and AA aero. Plus your situation looks like its different from mine.
Over all, I tend to think sealing around the lid wont be necessary if its a standard Home Depot tote with a big over hang on the lid. If it just has a small lip, or the lid sits very unevenly, then I would seal it.
The things you need to juggle can be tricky to balance out.
For one, you absolutely must have fresh air in the rez for the roots. If there is no air circulation, everything will turn anaerobic and the roots will just rot. BUT - if you have any kind of draft or blowing air, your mist gets dispersed and the roots dry out or you waste a lot of nutes trying to keep them wet enough.
Thats why I mentioned the net pots having hydroton. Air moves through it fairly easy unless its choked with roots. It you dont top drip, the roots should mostly grow down seeking the water. Late in the grow, they may tend to get blocked up though depending on how long you run.
You might see if you can think of some other ways to allow for fresh air while keeping light and drafts out. Maybe a hole with a fabric covering and a light shield?
My situation is different. I grow in a small tent that sits on top of my work bench. The plants sit in the grow space in the tent above the bench top. The root chamber is suspended below the bench top and insulated from the heat thats in the tent by a layer of 1" thick foam.
The root chamber is actually two fabric pots stitched together to form a 55 gallon chamber - 24" in diameter x 29" tall.
The fabric serves two critical purposes. First, it breaths, allowing fresh air into the root zone, but with no drafts. Air can filter through the fabric, but not blow through it enough to disrupt the mist in the chamber.
The second thing is that as the fabric gets wet from the mist, that water evaporates off the fabric cooling it down. That evaporative cooling drops the root chamber temp anywhere from 7 to 10 deg F below what ever the ambient temp is. It works just like a swamp cooler. It works so well, I actually have to heat the root chamber much of the time.
To keep light out, and to help with heating, the entire area under the bench where the root chamber is, is surrounded by another layer of that 1" foam. A small heater sits in between the foam and the fabric and kicks on where ever the root chamber temp drops below 68F.
Im not sure how well my system might work in your situation though. It looks like your totes are in a larger room sitting on the floor? You might think about that though. I suspect you will have trouble keeping the root chamber temps low enough with the totes right out in the grow space. Thats also a common problem with most DWC, and ebb/flow, and other hydro setups that have the rez in the grow space.
Plus, my setup would be difficult to do for anything other than drain to waste, which would be hard to do with foggers in a rez. You would have to get creative. 
It just occured to me that one thing I said above needs some revision. I said you could probably run at much lower EC levels, but that will only be the case as long as the roots are above the water level. Once they hit the water, they are going to want normal hydro/DWC type EC levels. At that point you will be running a hybrid system.
Hmmm. It occurs to me you may also run into issues with the roots over running your foggers, so you might want to plan for a way to easily move them if necessary.
Even with my 55 gallon chamber, the roots have over grown the space on every single grow - even with just one single plant! Roots always go for the water source and will fill the space given enough grow time.
