Is it permissible to urinate in the shower, and perform ablution in a place designated for washing hands inside the bathroom, while reciting the supplications for entering the lavatory and for ablution?

It is permissible to utter the prescribed remembrance (dhikr) when relieving oneself in a showering area, provided it is not a toilet, and similarly for the remembrances recited during ablution. However, if the showering area is within the same space as the toilet, then mentioning Allah therein is disliked. In such a case, it is better to say "Bismillah" before entering and to perform the remembrance after exiting. Urinating in a showering area is disliked if there is no drain for the urine. But if there is a drain, then there is no dislike, especially when there is a need and difficulty in urinating in the toilet.