Our most popular general use product is called Talstar. It is labeled for over 75 different pests, can be used indoors and outdoors, and is pet safe if used as directed.
Most insecticides like Talstar P and herbicides such as 2,4-D, are able to be tank mixed together, just make sure you follow each product label instructions for "tank mixing" instructions.
Talstar P Professional Insecticide can be sprayed or misted but not fogged. For most application it will be 1 oz per gallon, but review the label for complete rates.
https://www.domyown.com/msds/Talstar_P_Professional_Insecticide_Label_2020.pdf
Talstar P (Talstar One) dries within approx. 2 hours. After that rain will not wash it away for at least 2-3 months. Irrigating after spraying will only wash the product away. Most recommended insecticide sprayers use a range of 30-150 PSI. Higher-range PSI will tend to break down the insecticide.
There is no real reason to mix both Talstar and Cyper. They kill pretty much the same insects. One or the other would most likely do the job.