I have the same trouble with pots in the sun. Every day is 95 here (or more,) and though the foliage will be really diggin' the sunshine, the roots can get way too hot. There can also be faux wilt in the heat of the afternoon, when plants wilt but the roots still have plenty of moisture. This can happen to potted plants much more easily than those in the ground. If your schedule will permit, check for thirst in the morning, for any plant during the heat of summer, try not to water when plant is overheated.
Using white pots can help, and/or finagling some shade for the pot but not the foliage. Basil is such an enjoyable plant, to grow and, of course, to eat. I would encourage you to try again if still interested.
Before it gets too hot, you could probably grow some lettuce, which likes cooler weather. Most edibles first make a flower, then the veggie, so might exacerbate your allergies, IDK.
For house plants, your options would be huge. If you look at the plants available in stores, you could make a note of what they have and come back here to ask if they should work. Just avoid those that need the most sun, like most cacti/succulents. Tropical, leafy entities from the shady understory of rain forests should love the conditions you have. I tend to fall in love with the pretty leaves on plants like Philodendron, Anthurium, Dieffenbachia, Syngonium, Aglaonema. If little trees are more your thing, there's Dracaenas, Scheffleras, Radermachera (China doll,) various Ficus. Sansevierias should do well too, they'll put up with about any location. Creepers like Tradescantias, Callisia, Gibasis, Ceropegia, various Hoyas should enjoy your balcony.
