I can not emphasize it enough, A Test Strategy is not a document! Whenever you test anything, you have a Strategy in your head. Most of your Strategy is in your head or in several heads spread over the team and will become clear when you discuss it. Writing things down, making mind maps or other visuals and models only helps you understand and communicate your Strategy better. The trick is to become aware of the Strategy in your head (mental model) and create insight and overview for yourself and your team and the stakeholders.
We have strategies for everything we do, even when we do not write them down or visualise them. Ask yourself, do I need to write it down? Why? To discuss it? To make it better? To communicate it? To report or for accountability purposes?
When you think of how to test a whole project or a separate user story, you think of different things but also about stuff that overlaps. I think it is a good idea to “split” your Strategy up and examine it from different angles: project, features, user stories, releases, etc. Using many perspectives and many models (see my answer for question 1 and question 2), will make your thinking better. Which does not mean it needs separate documents or in some cases even a document at all (a Strategy is not a document remember?).