What is the ruling on water droplets falling on my clothes and body from the hanging laundry of Christian neighbors? Do I have to change my clothes and wash what the water touched, or is the original ruling for this water that it is pure?

The default status of droplets falling on a person and their clothes is purity. It is not obligatory to wash what touches the body or clothes from them, even if the water's color has changed; because the default is that something remains as it was until its change becomes evident.