Is there any care required? I.e. Trimming in fall? Many thanks!!!
It seems that this plant is well adapted, and does not require much extra care. You may prune it if you would like, however.
Here is an article about them: https://www.gardeningknowhow.com/ornamental/trees/tamarix/tamarix-information.htm
I have a tamarix tree hedge behind my house in the California desert. Is there anything that can grow under tamarix or near tamarix?
You can probably grow anything near the Tamarix that thrives in desert conditions, but be prepared to have your plants crowded out and taken over by the Tamarix, which is extremely invasive and can be difficult to control. Here's a link for a series of plants that grow well in the desert, as well as an article about Tamarix plants.