What should I do?

First, upgrade your facilities to prepare a good learning environment for the cats. That's the first step! If you also proceed with hiring and training professors, you should be able to clear stages efficiently, right? Don't rush, take it one step at a time, and aim to build a magnificent magic school, Hou!

I think it would have been better if all items (like desks and beds), including the dorm, were fully prepared. You don't need to level up each item. I don't quite remember if all the teachers were necessary, so it would be great if someone could clarify.

I'm not entirely sure, but... it seemed to clear once I met the conditions to move on to Stage 2. I unlocked all the furniture like the lodgings (though I haven't finished upgrading them), but I didn't have all the teachers. Just for your reference...