16"s will fit, there's a Corolla spec with this size steel wheels. But wheel size is only one factor affecting comfort, before you go changing. I have a GR Sport on 18"s, it's sportyish but I wouldn't have said a harsh ride, is pretty well controlled. Can you try first?
What year/model of Corolla? Unless it's a GR, 16" wheels will be plentiful, cheap and no worries with fitment.