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Yes, you just need to select a single trunk. These links will help you.

https://www.gardeningknowhow.com/ornamental/trees/desert-willow/pruning-desert-willows.htm
https://www.gardeningknowhow.com/ornamental/trees/desert-willow/caring-for-desert-willow-trees.htm
https://plants.usda.gov/plantguide/pdf/pg_chli2.pdf

I would suspect that your heat/drought conditions and appropriate deep watering should be your first consideration.

https://www.gardeningknowhow.com/ornamental/trees/desert-willow/caring-for-desert-willow-trees.htm

Desert willow trees are know to be relatively resistant to pests and diseases, but there are some possibilities for disorders:

Invertebrates:
Aphids

Diseases:
Verticillium wilt

Environmental disorders:
Mineral deficiencies
Nutrient and mineral excesses
Poor water management

Weeds
Vertebrates: gophers

It is not possible for us to give a definitive diagnosis without more information on growing conditons and management. You may need a site visit from a local consulting arborist or university extension service advisor.
https://plantclinic.tamu.edu/

I do not recommend planting anything near water lines, although some things don't have invasive roots. I would make sure to keep it at least 5 feet away from water lines just to be safe.

These are considered invasive in many areas, and won't require feeding. The soil nutrients, even if very low, will be enough. This article will give you more information about these: https://www.gardeningknowhow.com/ornamental/trees/desert-willow/caring-for-desert-willow-trees.htm

It won't "heal", but it may hang in there without completely splitting and falling. It depends on the severity of the split and how much end-weight and movement there is on the branch. Here are some options:
Prune the end of the branch to moderately reduce weight and wind resistance. This will prevent the weight and movement worsening the split.
If there is not a great deal of separation and twisting preventing it, drill through both sides of the split and install a "through-bolt" or two with washers and nuts to prevent further splitting.
Install a vertical prop, or A-frame support, from a solid footing at soil grade, to artificially support the split branch and prevent further splitting.
Remove the split branch at the point of bifurcation with the parent limb. Don't cut into the parent limb to achieve this.
If the parent limb has sustained a severe wound from the split, consider one or more of the above.

The answer is probably. If the tree is well balanced and has roots all around the base then one or two being trimmed shouldn't matter. Just make sure the tree is watered if there is a shortage of rain or it may wilt a little in part due to the reduced root volume.
Steve
http://www.treestylearb.com