Yes, the European Union should adopt a universal written Constitution as doing such would ensure that human rights were respected and followed in all of its member nation-states. To have such an equalizing document would also ensure that any disputes between member nation-states would be handled justly and fairly for all involved.
Yes, the EU should adopt a universal written constitution, because it would give the EU direction. A written constitution will give the EU direction and purpose, which it sometimes lacks. It would also give member states a chance to thoroughly discuss what exactly they want the EU to do, which would increase unity.