- "...should the student work on open problems...?"
I think that working on open problems is fine—even exciting—if you follow Pólya's advice in How to Solve It:
"If you can't solve a problem, then there is an easier problem you can solve: Find it."
Just as one example, it is unknown whether or not it is decidable if a given single polygonal tile can tile the plane. It is unknown even if the tile is a polyomino. I supervised a very nice undergraduate exploration of this question for specific polyominoes.
If you simplify the question enough, you can solve it. Then generalize until you cannot solve it. And so on, oscillating above and below.