You can do wudu naked because uncovering the `awrah or wearing shorts does not invalidate wudu.

There is no proof that the Prophet (peace and blessings of Allah be upon him) said any du`a during wudu whilst washing or wiping his limbs, etc.

If, after praying, you find on the parts that must be washed in Wudu something that prevents water from reaching them, then you must remove it and repeat your Wudu and prayer. As for cutting nails, it does not break your Wudu.