Is it forbidden to bathe completely naked in the bathroom, and is it obligatory to cover one's nakedness during bathing?

It is permissible to bathe naked in private, and the proof for this is the bathing of Ayyub and Musa (peace be upon them) naked in private, and Allah Almighty did not rebuke them for it. Concealing oneself with a loincloth or similar is better than being exposed, but exposure is permissible as long as there is a need for it.