For less elaborated fence panels you use hangers to support the horizontal 2x4 boards but with this desing that is not possible. These are pre made panels
The hanger you have pictured is not correct for the panel you have pictured as the 2x4 is not oriented correctly for them.
If you are not comfortable using just deck screws you could put an L bracket on the bottom of the panel to carry the weight of it and use a 1/4" lag bolt or carriage bolt at the top for lateral or shear strength
The frame is made of 2x4s so yes you use scres to attach it straight to the post Makes no sense to me to have that frame (at least the side edges) when you can use two hangers at the end of each traversal 2x4