With the small pots you have, use one of the mixes with the water retaining additive. Shading the pot from the direct sun will help. You can try Styrofoam like in a minnow bucket, and poke in drainage holes if you use one. A layer of sphagnum on the surface of the soil will help the soil stay cool. Roots in pots do get cooked to death in the direct sun.
Now for the flowers. You could try some border dahlias. The tubers they have help them survive the soil drying out, and they flower almost all the time if you trim the old flowers off. Black Eyed Susan, Cone Flower, and Coreopsis are very hardy heat and dry tolerant plants. You can grow almost anything that is considered a strawflower. I think you could grow these and less hardy plants if you address all the issues I brought up.
I (of the brown thumb) have had good luck with several varieties of coleus. They don’t flower, but they come in a zillion colors and patterns, and depending on the type, they do very well in our hot and sunny backyard.
May I join the petunia contingent? The purple wave variety survive in municicpal hanging baskets in full sun in Georgia where the temperatures are often obscene, looking beautiful all the while. I got some Kentucky heirloom petunia seeds from JL Hudson and they have a nice variety of color.
Keep in mind that window box containers are generally made with very good drainage. Very Good.
If the plants are out in the hot, direct sun, in containers, it will be almost impossible to keep enough water on them, in my experience. I share your climate and have suffered a similar problem.
Someone recommended this to me for my hanging baskets and it really worked wonderfully. Line the container or window box with a large plastic bag, poking only one or two small drainage holes in it. Now, when you water much, much more of the water will stay in the container and help to save the plants from getting too dry.
It proved just the thing for my hanging baskets which were suffering terribly, now they truly thrive, though they are still in a very sunny location.
Of course, choosing sun loving plants will go a long way to helping them thrive regardless. Just check at the garden centre, they’ll have a special section I’d bet.
Okay! Thanks for all the advice. I popped into a garden centre yesterday and picked out a few things based on the suggestions here and the suggestions of the staff there. I can’t remember the names of all of them right now except for two - Brass Buttons and Creeping Thyme. I’ll look at the rest again when I get home.