Is it okay to ask a journal for review reports as they are available?
There is nothing wrong with writing them a polite letter asking what the status of the submission is and whether they have an estimate for when it will be done.


  The problem now is what if the reviewers ask for additional experiments to be performed, but the student who actually knows how to perform them has already left the lab?


This is really a separate question, but if that happens, typically your options are:


Convince the editor to let you skip those additional experiments
Try to get the student to come back and do them
Have someone else do them, hopefully with the benefit of the original student's notes and advice



  Would it be okay to ask the editor to release any referee reports already completed while they wait for a final decision? 


I doubt they would agree to this. For one, it is uncommon to do this so most editors would refuse for the sake of being consistent with their policy. Also, sometimes the reviews can influence each other, for instance the editor might wish to investigate a disagreement between two reviewers, and the reviewer that finished early might want to change their review. Worse yet, it may be that when the late review comes back, the editor will decide to reject you outright. If you have already done some revision work at that point, it becomes an awkward situation. As an editor, I would never agree to this, simply to avoid weird situations like this.