Yes. You are correct. You will only need to configure the ports with hosts connected (Servers/PCs etc..) as edge ports and not the trunk links between switches.
Please let us know if you have any other questions.
You should always specify a root bridge, otherwise the system will determine a root bridge based off of MAC address and that can be unpredictable. all your nodes will block or forward based upon path to that root bridge, so usually it is place pretty centrally in the environment.
Please bear in mind that if the user ports of HP switch are not configured for edge port (or PortFast in Cisco terms) then each time a PC disconnects/connects you will get a Topology change. So you should configure edge ports on both switches' user ports.