University lodging isn’t as simple as it might sound. Bologna is feasible. The university there sprawls and has housing. But in Florence and Rome students attend day and night courses and live at home. There are no dorms. Your plan, while intriguing, doesn’t lend itself to simple solutions. The overall dilemma in Italy is housing — not enough of it — which is why hotel costs are so high. The demand is there. You could stay in religious spots. Those exist. Opting for a pensione might be wiser.