1) If you touch a dead dog, then you will be unclean until evening.
2) If you pick up a dead dog, then you must wash your clothes, and then you will be unclean until evening.
The reasoning here starts with the idea that the demons inside a dog are adapted to occupying an animal but not yet adapted to occupying a person. Such demons cannot transition from an animal interior to a human interior easily. They can accomplish the transition, however, if they first establish an intermediate base on a person's clothing. After they have become acclimatized to occupying on a person's clothing, then they can enter the person himself if an opportunity occurs -- for example, if the man ejaculates. The act of ejaculation opens the man's body momentarily, which is long enough for the demon to move from the clothing into the man.
When a person merely touches a dead dog, the implied understanding is that the only contact was between the dead dog and the person's skin -- and that there was no contact between the dead dog and the person's clothing. A demon cannot move from the dead dog onto the person's fingers; the demon can move only onto the person's clothing.
When a person picks up a dead dog, then the implied understanding is that the dead dog touched the person's clothing, and so a demon was able to move from the dead dog onto the clothing, which then might serve as a base that will enable the demon later to move into the person.
Somehow, the setting of the sun causes the person to become significantly less vulnerable. The person's sweating skin during the daylight's heat caused some vulnerability to entrance by a demon, because sweat is a bodily discharge, somewhat like semen or menstrual blood. After the sun sets, and the person's body stops sweating profusely, then the person's skin becomes a solid wall that demons cannot penetrate.
So, if a person touches a dead dog with his hands (not with any clothing), then his sweaty hands make him vulnerable until the sun sets and the temperature cools and his hands stop sweating. Any demons that came from the dead dog and that are able to wait nearby for an opportunity to enter the person will have to depart when the sun sets.
If a demon from a dead dog managed to get onto the person's clothing, then washing the clothing with water moves the demon off of the clothing, and the demon will have to depart when the sun sets.
Another explanation for the effect of sweat is a water that washes the skin adequately for situations such as a person's skin touching a dead dog. During the daylight's heat, the skin sweats enough to prevent such a demon moving from a dead dog and attaching itself to a person's skin. Since the person's clothing does not sweat, the demon can attach itself to the clothing.
The important concept I want to communicate here is that in the case of contact with a dead dog, the person's body does not have to be washed with water. Only the person's clothing must be washed with water, if the clothing touched the dead dog.
Dead Mouse: A mouse is unclean because of two reasons -- a mouse creeps (walks on legs so short that the abdomen often touches the ground) and swarms (moves quickly in groups). A demon can move from the ground up into the animal not only through the mouse's paws but also through the abdomen, and the latter route enables more powerful demons to enter. The swarming is an indication that all the swarm's animals are possessed by demons. So, even though a mouse is much smaller than a dog, a mouse might be occupied by a demon that is much larger and more powerful than any demon in a dog. Leviticus describes the greater danger:
Should any of these creatures fall into a clay vessel, everything in it becomes unclean, and the vessel itself you must break. Any solid food that was in contact with water, and any liquid that men drink, in any such vessel become unclean.
Any object on which one of their dead bodies falls, becomes unclean; if it is an oven or a jar-stand, this must be broken to pieces; they are unclean and shall be treated as unclean by you.
The next new step in the reasoning is the difference between still water and moving water. If a dead mouse is in clay vessel full of water, then the water was still, without movement, long enough that a demon is able to move out of the dead mouse and establish a temporary base in the still water, from which the demon then moves into the vessel itself. Therefore, the vessel is possessed by the demon and becomes unclean and must be destroyed.
If a dead mouse is in a spring or a cistern, however, the water always is moving enough that the demon cannot move out of the dead mouse and establish a base in the water. The demon stays in the dead mouse, because a demon of the kind that occupies a mouse is a demon that cannot transfer itself into moving water. If a person picks the dead mouse out of the moving water, then the mouse still is unclean -- the moving water has not cleaned the demons out of the dead mouse -- and so the person will remain unclean until evening.
The same reason applies to a situation where a dead mouse is in some grain. The demon cannot move out of the dead mouse into dry grain, but can move into moist grain, because this water is still.
In this context, the cleaning of wood, cloth, leather or goat hair and other such items in water necessarily involves stirring the water with the items for a significant length of time and then letting the items lie in still water for a significant length of time. Because the water was stirred first, the demon was frightened and wanted to escape, and then when the water was still long enough that the demon was able to escape, the demon did so. The demon was too frightened to stay in the water's container. When the sun set, the demons escaped.
Summary: When a modern person thinks about using water to wash a dirty object, we understand the effect as the water's movement removing the dirt from the object and carrying the dirt farther away from the item. Certainly an ancient person understood that effect likewise.
The ancient person, however, understood also another effect. Usually, a demon inside an object could not move from the object into moving water. Furthermore, the demon was frightened by the moving water that surrounded the demon, and so the demon would escape from the object if an opportunity occurred while the demon still was frightened.
For example, if a person washed himself in running water, such as in a river, then a demon inside the person was not washed away by the running water. Rather, the demon was frightened by the running water, and so as soon as the water stopped flowing around the person, then the frightened demon might escape from the person. The running water did not wash the demon out from the person; rather the running water frightened the demon out from the person.