There is no best way for a robot to intake water. It depends on if you want your robot to feel and act like a human, or if you are just using the shape. The position of your cooling system and how much liquid it will require.
If you want it to be and feel human, then you would use the mouth and throat system. Putting water anywhere else would give away the human feel.
If you want it to be efficient, then you would simply have an intake area they would use to pump it into themselves. Most likely in the chest or shoulder areas as the water will flow down into whatever container is holding it. It could also store it in its head, but you will probably want real sensors that look like eyes there, so it would be a bad idea. Too low (aka pelvis or legs) and you have to pump the water up to critical components, but you already need to do that to reach the head. You would also need to consider how much that part can deform or moves (stomach area might be bad).
@Cognisant mentioned water sloshing around, and if you are evaporating water, this will always be a problem as you can't constantly fill up your water tank everywhere you go.
I don't think your traditional sweating idea would be a good way to cool any robot. Sweating works when the water evaporates and takes away energy hence cooling us. However you will require some airflow, something to transfer the heat to the "skin" and a mechanism to ensure water overflow and affect your electronics.
I suggest, rather than sweating, you use special heat vents to expel the heat. Water is used to transfer heat energy from electrical components to a set of heat sinks and fans which transfer the heat to the air. The water stays inside a set of pipes and pumps and you really don't need to refill it. If the heat sink is cool, nothing happens (in a sweating situation, the water will just stay on the skin), if its hot, the energy is transferred quickly from the components to the heat vent and cooled using fans, not how breezy it is outside. In this case you can also use oils or other special liquids to replace water.
The final way, would be to have all your electronics submerged in a special liquid. I'm not sure what its called, but there are videos online of computers which use this. Its expensive as hell and would allow you to submerge all your electronics and cool yourself using the skin to transfer the heat to the air around you.