Do I have to be logged on as a user with root authorization to start or stop SAP Host Agent?

It is not neccessary that you are logged on as a user with root authorization to start or stop SAP Host Agent. If you are logged on as user adm, you can control SAP Host Agent using the hostexecstart executable. 

However, it is not possible to stop SAP Host Agent without being logged on as a user with root authorization.