Reason for no header spanning this load bearing wall?
Why is there no header for the full height doorways?


I can think of two possibilities:


The wall is structurally unsound due to the missing support and if so, you should see some signs of that. (i.e. sagging, drywall cracking, etc.)
The wall is designed such that the unsupported span load is carried by the support on either side of the doorway opening.


Number 2 is most likely the case in my opinion.


  Do I need to add a header that spans the width of that area, or would removing those 2 studs in the middle be fine regardless?


That's a question for a structural engineer.  The loading from whatever this wall is supporting needs to be analyzed and an appropriate structure to transfer the load needs to be designed.  There is in no way adequate information in this question posting to provide an answer and, in fact, you will likely need engineer-stamped plans for your building permit application.