You should be able to grow all those crops hydroponically, except that blueberries might prefer a more acidic PH than mint, cilantro, and strawberries. Try growing the blueberries in a separate setup, but experiment with a couple of plants by all means in your current planned rig.
Your setup should be good in June, the problem with hydroponics in high heat, 80s plus, is that the water and fertilizer will get too hot and this can interfere with the metabolic of the plants. You can overcome this by putting the reservoir someplace cooler, i.e., inside or put it in the ground, or at least shade it. Plus try not to use clear containers for the reservoir because it will promote algae growth, especially in the hot Texas sun, you can always paint it white, or use contact paper. You will also need to watch that the chemical mixture PH and strength stay in the useful range, not requiring daily checks, but after a month you want to check it regularly. Keep a log notebook of the numbers by date, how your rig is doing, and the size of the plants. Bigger plants gobble up nutrients faster.
Start with seeds or small plants when possible; larger plants tend to die back initially, in this case, bigger is not better. As the Summer progresses, you might limit the amount of sun, but you might not need too. Remember you can always move the rig, so it gets shade at certain times, or get some shade cloth to protect it from the noonday sun.
Also, pull a couple of plant pots out to inspect the roots once in a while to catch any problems early.
As for other types of plants, all the common herbs, such as oregano, parsley, thyme, basil, chives, rosemary, savory, etc. can grow hydroponically.