Another quick question then. Is there a mulch that would discourage rodents from digging in the garden (my wife thinks it's a skunk..posums and racoons are common here too)?

That does complicate things, I have more earthworms in the areas mulched with leaves, and I think the larger critters dig them up. I haven't had digging in the areas mulched with several inches of wood chips, although the crows poke around in them and make a mess.

picture of wood chips

I thought melons and corn thrived in heat but this site doesn't mention them.

http://blackgold.bz/hot-summer-vegetables-that-beat-the-heat/

Consider a fig tree. They can take a lot of heat. For that narrow space a Black Jack (dwarf black fig) might work well.

Pomegrantes also take ferocious heat in stride; there is a middling-to-small size called Chico (don't get a "dwarf" pomegranate because that is a knee-high non-fruiting shrub).

Consider very large terracotta pots filled with palm/cactus soil for herbs. Most kinds of herbs will flourish (except basil wants richer soil).

If you just want an ornamental tree, try mediterranean fan palm. They take years to mature at about 12-15 feet, and even then they stay in bounds.

For tree mulch try cobblestones or a thick bed of river stones. These let water through, keep soil cool, and are not easily dug through by animals. The downsides to stone mulch are: no soil enrichment through decay, and the only way to get litter out is with an annoying leaf-blower, for which your neighbors will resent you.

Good luck finding the right plants. It is a pretty space.

The mass of bare branches to the left of our fence is our neighbor's huge fig tree. I planted (from back to street) a apricot, peach, dwarf mandarin (not visible in the photo) and a dwarf plum. We'll see how they do. The green in the photo is buckwheat which I was told to plant in the off season and then till into the soil once it flowers which I just did. Now I'm ready to plant.

Carrots? Radishes? Beans?

Any heat loving edible should do ok there - maybe not producing as much as it would in full sun but they should still produce. Beans (pole even up the fence), peppers (I've grown some in more shade than you have there and as long as it's hot/dry enough they've done just fine), maybe tomatoes depending on how hot HOT is - they won't die but they won't set fruit if it's too hot. Squash if you do fast producing squash and just keep replacing it as needed from heat/disease (that's what I do here).

We get pretty hot here in the summer and I avoid planting carrots during the hot months as they just don't do well and tend to not taste as good. Radishes tend to bolt in the heat before they get to any decent size.

Anyway, I usually put peppers, beans and tomatoes in my hottest areas...peppers being first choice as they really enjoy heat. Mulch, mulch, mulch as well (keeps their roots cooler and damper which will help with heat resistance).

Whoops. Just planted radishes. Hopefully they'll ripen before the heat. We're hoping for a "Miracle March" around here- some much needed rain. So far in a So Cal the El Niño has been a bust.