Is it obligatory for a man and a woman to conceal themselves from each other when urinating, and can Aisha’s hadith ("The Prophet never urinated standing up") be used to infer that it is not a condition for a man to conceal himself from his wife?

Neither a man nor a woman is obliged to cover from one another, due to the Prophet's (peace be upon him) saying: "Guard your 'awrah except from your wife or what your right hand possesses." However, uncovering the 'awrah without need contradicts modesty. As for Aisha's hadith (definition) about the Prophet (peace be upon him) urinating while standing, it does not indicate the uncovering of the 'awrah, and it has been authentically established that he did urinate while standing.