When our contractor graded the ground near our Jerusalem thorn trees, apparently he spread a lot of seeds, as we now have about 20 of them sprouting up on the as-yet unlandscaped ground. I would like to get some pots and transplant them. I am wondering how deep I need to shovel at this point (they are about 18-24" tall), and also wondering if there are any tips for what type of additional soil should be there and any specific tips on care. Thanks
Most likely the seeds were already in the soil, lying dormant, and the grading of the soil is what enabled them to germinate. You may notice more seedlings starting to grow if the soil is disturbed again, say for planting, because copious amounts of seed are produced each year.
Hard to say how deep the roots might be, the only way to find out is to try to dig one out, but please be sure this is not listed as an invasive species where you are. In Australia, this plant is considered the number one invasive species and efforts have been made to try to eradicate it. Further info here https://en.m.wikipedia.org/wiki/Parkinsonia_aculeata