Anxiety / meltdowns: We got some over the ear passive noise cancelling headphones for my daughter (who is moderately autistic and what you wrote would be a perfect description of her as well). They're lifesavers in public places like restaurants and stores. We have her on a pretty regimented schedule to try to minimize unexpected transitions. This helps her focus and really helps her sleep. What kind of therapists is she working with? Our daughter is diagnosed with moderate ASD and has therapy for ABA, Speech, and OT. She's 4.5 years old. We found a "sensory room" in the house is an incredible place for her to self sooth and calm down.
Sleeping: not a ton of good options. We frequently are up with her at odd hours. I'll bring her upstairs and we will watch Cocomelon at 3am for 30-45 minutes. I'll spend time sitting and reading with her as well. We often try to "wear her out" before bed by just playing a lot for a few hours before or after dinner. The hatch[1] was recommended by her ABA therapist, but we've not yet purchased it.