I'm not familiar with Arizona weather - I live the opposite end of the US. I expect it is hot and dry? But your box is in mostly shade? That is tough since all the herbs you list prefer full sun. Spearmint, parsley, and chives would be the most sensitive to hot, dry conditions. They prefer cooler, moister conditions and can provide challenges for southern gardeners.

Thyme, sage, and rosemary are Mediterranean herbs and prefer lean, dry soils with plenty of sun.

FataMorgana

The dimensions are actually very small for more than a couple of plants. You don't say the depth but I assume it's also 6"? Rosemary in mild climates will form a large bush so I don't think it's a candidate for a small box like that.

Sizewise it could probably accommodate 2 thymes or 1 sage or 2/3 parsleys or 2/3 chive clumps or 2 oreganos or a couple of mint roots. Parsley, chives and mint, as Fata says, need cool moist conditions. The others are Mediterranean lovers of sharp drainage but I would still be worried about that small volume of soil getting too hot, even for the Med herbs.

Sorry not to be more encouraging. By all means give it a go - there's not much to lose by trying.

One mature parsley plant would fit into that box nicely - assuming that the box is deep enough to accommodate parsley's tap-root. Chives would be OK, too. Try garlic chives, which are tougher than onion chives.

I think perhaps something like thyme would do OK in it - one plant will spread out and cover it, spilling over the sides as well.

Why not use the box as a seedling tray? It's really too small for much else, except as a very temporary measure.

Another idea would be to plant some salad greens in it - lettuce of assorted types. These are fast growing, you can pick as they grow, and you can keep on re-seeding from time to time to have an almost permanent crop. Draining needs to be good, potting mix needs to be rich, they need regular watering, but they can take the heat and sun. You can pack a lot of lettuce into a small space, if you take care to harvest 'in between' ones early.