Sounds like you have a big project ahead of you! So I will help you out with a few of the questions.
Let's start with the soil cost: Mel's mix is expensive, and it is not the only option out there, but it is a very good one in my opinion. Honestly, soil costs are going to be your largest investment no matter what. You can probably get the wood donated from a construction site. You will find that many people on this site are against using any soil in container gardening. The idea is that soil compacts too much and creates a muddy drainage issue when used in a container. Mel's mix is essentially a soiless mix as there is no "dirt" in the recipe. Now with that being said, I do have a box that I used 90% garden soil in. The same kind of soil that is 4-5 bucks at home depot. I just added some peat moss to go along with it. You should probably do a quick search of soil mixes on this forum and will find out what does work and what doesn't. I have heard a lot of warnings against using solely top soil due the reasons I listed earlier. I am sure you could amend it with compost and vermiculite, but then you are back to the same cost issue!

Now as far as plants go: Think spacing and think ease of growing. What I mean by this is that some of the plants you have mentioned need a lot of space/care.

Easiest things to grow:
Lettuce, Peppers, Tomatoes, Beans (bush), herbs, spinach.

Things to consider:
Corn: Heavy feeder (needs a lot of fertilizer) and needs a lot of plants to successfully open pollinate. Unless you plan on manually pollinating the silks (easy to do, but may be tough considering your situation) you should probably avoid unless you plan on using an entire 4X4 box for solely corn. I actually grow only 8 stocks at a time and self pollinate, so it is possible and may be educating for the children, but corn is also very big so pick a smaller variety.
Squash/Cucumbers/Zuchinni: These grow very large and one plant can easily take over 16 sq feet. You can do these, but you may want to take them vertical. They are very large plants once they get going, but are also very fun to grow.
Strawberries: Easy and fun! You should really do a couple squares of these for the girls, but understand that strawberry plants do not really start to produce big fruit until their second year and you need to clip the runners that grow from them, so they require some pruning. You also want to pinch buds for the first month or so in order to promote rooting.

Egg plants, peppers and Tomatoes: Start from seedlings if possible. It will be a bit late in the season to start from seeds and these grow slowly. Eggplant is the more difficult of the three, but you will be fine. There are two types of tomatoes, Indeterminate and determinate. The determinate are probably easiest to grow as you really do not need to prune them and you can just cage them.

Beans: Bush beans do not need a trellis to grow, so if you do not plan on having trellises then only get the bush kind. Pole beans need a pole or trellis to grow.

You can plant bush beans along any of your heavy feeders as they actually pull nitrogen from the air and put it in the earth for your feeders to eat. you can also try a "three sisters" box (corn, squash, and pole beans). This was the way the Iroquois Indians planted theirs. Search this forum for directions on how to do this. Like I mentioned earlier you may have to manually pollinate the corn, but this can be done by shaking the stalks if you have enough in the container.

Thank you so much for the reply. I just now noticed it. About soil, my local Lowes has Garden Time. Is this good enough? It's expensive, but we managed to get Home Depot to donate all other needed supplies for this project, so I think I can justify the expense now. Yay!!!

Since we won't be managing the garden after its planted, I just need to go with the very easiest of items to plant. Only stuff they will have success with. Isn't it too late for lettuces and spinach though?

I took copious notes on everything you suggested. I'm not going to do corn, sounds like too much work for our situation. I guess now, I just need to figure out spacing and what to plant with what.

If we want to plant some items as seeds, should one whole box be just seeds and another be all seedlings? Should we plant items with the same harvest time together in the same box or does it matter?

Sorry for the long delay as I just stumbled upon your post again. So to your questions:
1.) Lettuce and Spinach: yes it may be a bit late now. You could possibly squeeze in some late season mixes under in the shade of some of your larger transplants.
2.) seeds: it is a little late, but you can still get by with some faster growing veggies. Beans grow fast so they would be fine for sure. Check your local planting schedule for more info. Just google your city's name and planting schedule.
3.) transplants for peppers and tomatoes at the very least! Peppers can be one per square foot, tomatoes may need a little more space if you do not plan to prune them. Give them at least two feet each in your situation. I grow one per square foot, but I prune them down to one vine and stake them.
4.) Oh and you soil should be fine. Maybe add a little compost to it.