I think the answers will pretty consistently be that you need to give each of these guys their own pot. Maybe you can get by with some two to a pot. When spinach and lettuce do well, one plant can fully occupy a pot that size. You might get three, maybe even four, radishes in one pot. I think carrots will be a problem, unless they're dwarf or you intend to pick them immature. Corn will likely not make at all, even one to such a small pot. (Corn leaves commonly do look like that. Lots of things nibble, some of them without doing too much harm. If it gets serious, deal with it with insecticide soap or Sevin, depending on your preferences.)

Now, if you leave such things as lettuce and spinach crowded like that, they will make some usable product, but it will be some leaves on small and stalky plants. They won't bush out. Ten spinach in individual pots will keep you steadily in spinach. Ten plants in one pot may not ever give you enough for a dinner party salad.

You kind of have to keep an eye on soil moisture. Leaf vegetables are nearly all water. They will do well when the soil at root level stays just moist enough to feel not dry all the time and doesn't experience wet-dry cycles, moist enough that you can tell there moisture in it. depending on the plants location and your climate, that might mean once a day watering, or it might mean constant wick watering.

these are all mini/baby/container specific varieties. For example the Envy soybeans only get to be 2 ft tall max. I got all my seeds from http://www.reimerseeds.com

Here is a list of the varieties/breeds(?) of plants I am growing:
Edamame- Envy Soybean
Carrot- Little fingers
Corn- Baby Corn
Radish- Sparkler
Spinach- Melody
Lettuce- Tom Thumb

Given this information, does the same advice hold true about how many in each pot?

One problem with corn is that you need a lot of plants to pollinate one another, like 20 or more. Did the seed company say you could grow their corn in a container?

it didnt NOT say you could grow it in containers. I bought it based on websites that said baby corn does well in containers.