Thermistor with this particular temperature behavior are commonly semiconductors. In a semi-conductor, there is an energy gap between the (filled) valence and the (empty) conduction band. At zero temperature, no charges are in the conduction band and the resistance should be infinite as the system behaves basically like an insulator.
If you turn on the temperature, some electrons will start to occupy the conduction band and thus contribute to conduction, lowering the resistivity.
Firstly, it would be better to use actual, accurately measured numbers than the human 'experience': the human body is a poor thermometer and the mind plays tricks on us.
But that hot water droplets lose heat and thus cool down in cooler air is an established fact and a consequence of the laws of thermodynamics.
Regarding your three first bullet points, despite some limitations you point out, those do not mean a hot droplet of water doesn't cool in air: it does and partly in accordance Newton's cooling law.
As regards work done by the droplet (overcoming the viscous drag), if anything that would lead to heat generation, not cooling (but the effect is truly minuscule).
Kinetic or potential energy of the droplet have no effect on the droplet's temperature. Temperature is simply a measure of the average speed of the molecules of the water and that is not affected by these energies. Spinning water in an ultra-centrifuge does not make its temperature rise, for instance.
You have however overlooked one major cause of heat loss: evaporation. Your shower 'steams up' because hot water evaporates and that costs energy, known as the Enthalpy of vaporisation.
Millions of tons of water are cooled this way everyday in power plants world wide: the cooling towers drop hottish water from the top of the towers and evaporative heat cools down the water (the evaporated water escapes as steam clouds).
If your shower has been in operation for a long time and the bathroom's temperature is equal to the shower head's water temperature and the air is saturated with water vapour, then no cooling of the shower water would take place.
Best Answer
Vigorous stirring is exactly what James Prescott Joule did in his experiments, so the answer is "yes", the temperature goes up. Now the subtlety that you have corrently noticed is that coherent kinetic energy of motion for the fluid is not the same as a temperature increase. The resolution is that viscosity eventually damps out the coherent motion and turns it into incoherent molecular motion, i.e heat. Viscous dissipation is present whenever parts of the fluid have different velocities, and this exactly what happens when fluid is stirred.