Thank you for the opportunity for me to delve into the theory of testing and Quality Assurance. As you seem to be mostly interested in the role of a test specialist in software realization projects, I will keep the scope of my answer in that general area. Please keep in mind that most here is written from my opinion and experience and less based on books.
First about quality
Before we can start testing we need to understand the concept of Quality. Quality can be defined as the distance between the actual object under test and the customer's expectation of the object. This means that you should be measuring how well the object fits the expectation to determine the quality. To do this, you should be aware of the expectation, which is often captured in requirements. Now you can see that when your requirements are not fitting the expectation, neither will the end result of everyone's efforts.
What is testing and why would you do it?
In my opinion, testing is more about measuring than it is about suggesting improvement. Everyone has a specialty and the Business Analyst should be very skilled in delivering great requirements and suggesting improvements. This pushes your point 5. out of the window.
As a tester, you should be clear, in every new project you work, what your assignment is.
Are you expected to add to great quality software? Be prepared to meet with the customer to find out his expectations.
Are you expected to add to a low defect count? Write loads of test cases based on the requirements and specification.
Find out what type of tester you are and make sure you fit in your team, you will do fine.