Yes, you can mix Harmonix with another insecticide like Cyzmic CS as long as the application rates on each product label are followed. You can apply both products to bushies and shrubs outdoors. You want to apply when you are not expecting any rain for 24 hours. After that point, rain will not affect either product.
Harmonix Insect Spray cannot be used on horses and contains botanical pyrethrin not permethrin. PyGanic Specialty contains the same active ingredient but does list it can be used on horses and other livestock.