I started a new garden in the high desert 5 years ago and faced similar issues.
For mulch, I go out and cut certain stands of wild plants in my neighborhood, mostly in June before they set seed. In my neighborhood it's been mainly wild iris, a
perennial artemisia, and perennial pepperweed Lepidium. So they are all small sticks or stems with some leafy parts. I scatter mulch lightly over seeds when I sow direct out in the sun, otherwise the sun scorches the seedlings or they never even germinate.
I water with a hose to the edge of each sunken bed, in turn. The top of the mulch doesn't often get wetted.
I don't find pest issues related to the mulch. I don't till, and try to leave the mulch on the beds permanently. The beds that had mulch the previous season, the soil is dark, soft, friable, and full of life. The beds that I removed the mulch from for some reason revert to pale desert soil within a couple of months. Some organic matter and skinny
roots will remain in the dry soil but it will be hydrophobic and difficult to water and recover. The difference is dramatic.