One thing I do not see is drainage in that container. Are there any holes in the container for drainage? If not, you will need to get a drill and put a few holes near the bottom, no matter what you plant.
Peppers, herbs, annuals are fine for that container.
Lettuce, spinach, broccoli are what is called "cool season" crops. They grow in the cool season of spring. Once it gets hot they are done. You might want to check those out.
Cucumbers are doable however they are going to need a trellis to grow up on. They get HUGE. 1 cucumber plant would take over the entire planter if you do not train it to go up a trellis.
Peas also needs a a support system for growing and I do recommend replanting those. You need at least a 12" depth and some bamboo poles for them to climb up on. The reason I say replant those is that their watering needs are heavy during the hot summer which will not blend with the other plants. I use those 5 gallon buckets, you get at Home Depot, Wal Mart, etc. You drill several holes in the bottom and ta-da, a cheap planter that also has a handle.
Celery, I have never grown but maturity takes a long time. They don't like it hot and also they will have demanding water needs during summer. I do recommend replanting those to their own space as well. Be careful with temps right now. Below 55 degrees for several days may force them to go to seed instead of growing. I will give you a link here as celery is very needy with requirements. Might be too late, but that will depend on when the heat hits you and how hot it gets.
http://garden.org/foodguide/browse/veggie/lettuce_planting/439
Fertilizing. You will need to fertilize the plants in the containers once it gets hot.