What is the ruling concerning clothes that have been splashed with a fine spray from the water used for rinsing the mouth, mixed with blood from brushing teeth with toothpaste, given the difficulty in pinpointing the location of the spray, and it might extend to the underwear?

If the water used for rinsing the mouth mixes with blood, it becomes impure, and whatever part of the body or clothes it touches must be purified. However, if you doubt whether the blood mixed with the mouthwash water, the water remains pure, because the default state is purity.