I like all you listed, I'm sure there are more favorites. One I'd suggest is kohlrabi, in my opinion ther grow great, are beautiful, and above all are delicious

If you are planting pole beans, the chain link fence will not be tall enough. My bean (rattlesnake beans) trellis is 8 feet tall and the beans grew up and over the trellis at the top. This picture was taken 6/27/14. The beans kept growing above the trellis and all tangled up at the top.

I'm late to comment but as already mentioned, the fence will not be tall enough for pole beans although you could easily construct a taller trellis. You could do bush beans if you don't want to bother with a trellis. The fence should work for melons, just be sure to support the fruit when they start to develop. Snap peas have both tall and dwarf varieties. The tall ones get 5-6 feet and need support, the dwarfs get about 18 inches so choose accordingly. Peppers and lettuce will do well. Tomato plants, if allowed to do their own thing, are huge and you will need to prune them a lot in order to grow them in a square foot garden. The plants you want are all fairly easy to grow but each has its own learning curve and its own potential pests/diseases.

As for planting, peas will be the first thing in the ground followed by lettuce. Both are usually planted prior to your average last frost date. Tomatoes and beans are frost sensitive and are planted after your average last frost date. Peppers and melons prefer the ground a little warmer so it would be best to wait a week or two after planting the tomatoes/beans.

Personally, I'd plant cucumbers rather than melons. They are grown almost the exact same way but cukes will give you a much higher yield and you won't have to wait nearly as long to harvest them.

Apologies for the jumbled mess in the first paragraph.

Rodney