In requirements gathering I've learnt to outline the objectives for the project as well as identify stakeholders and the stakeholder needs (along with the users needs, which is off-topic for this question). Are project objectives and stakeholder needs one and the same thing? For example, for a website:
Project Objectives are typically higher level than individual Stakeholder Requirements. They usually come first, created up-front in the Project Initiation phase before you've delved in to individual Stakeholder Requirements.
Stakeholder Requirements are lower level and more specific that Project Objectives. They'll carefully define the needs of the various stakeholders. Remember that these can include Functional (eg gather feedback from users) and Nonfunctional (eg responsive design) requirements.
You'll find throughout the progression of a Project that many "outputs" are re-workings of previous "outputs" at either a lower level of detail, or with more clarity and certainty than before. This is because as a Project progresses you are continually gathering more information that help define the details of the project itself. You start broad (vision / objectives) and end specific (requirements / solutions).
In simple words, stakeholder requirements are generally part of the project objectives and considered part of stakeholder management. There may be many other requirements or objectives of the project as well. In some cases, requirements may not necessarily be an objective but may just be a part/input/ecosystem of a project wbs or plan. That should answer the 4 points you mentioned.
There should be no "versus" when you think about the relationship between these two constructs. Stakeholder wants and needs drive the reason for a project's existence. Both have varying degrees of abstraction--from the high level to the low level--and a requirement should be linked to a stakeholder want or need and vice versa.
