Excellent question!  Think of Service/Case/Task and Lifecycle event as tools in your toolbox. There are may ways you can solve the use case you described using these tools.  There is no one "right way" to do it.
At a minimum you must define the HR Service within a COE. That will result in the ability to create a case. All the work you describe could happen there but it would probably be confusing to keep passing the one case between the different groups.
Therefore you could use the HR Template to pre define tasks OR have HR Workers create Tasks as needed to assign out work for the various steps. This would allow for parallel efforts to happen where possible and create a good "paper trail" of who is doing work and what the status is of that task.
If all the different steps you described above can at times be triggered/requested separately, instead of tasks you will want to create an HR Service for each one. If they are Services, you can still trigger them from Templates or Create Task (HR Task Type = HR Service).
But if you have all of these Services that are strung together in a consistent/repeatable way, that is were Lifecycle Events allow you to package up these Services in Action sets which are triggered by dates, other actions or the completion of prior Tasks, Services, or Cases.  Let me know if you need further clarity.