What is the ruling on someone who was performing ablution (wudu') and touched his hair with an impurity, so he washed it, and then kept his hand under the tap to maintain its wetness? And does this invalidate the order and sequence (tartib and muwalat)?

The questioner must strive against whispers (waswas). As for the specific question, there is no disruption of order or continuity (muwalat) in wiping the head after washing the hands, even if the separation time is prolonged due to removing impurity from the head. The Shafi'is and Hanafis consider continuity (muwalat) a sunnah (recommended) and not obligatory. It is permissible for someone afflicted by whispers (waswas) to adopt the easiest opinions.