Why isn't water running faster hotter?
The water gets colder the longer you run it (in the UK at least) because the water mains pipes buried in the ground are colder than the ones in your house, so sadly this isn't evidence for any fundamental physical effect.
In principle any fluid flowing in a pipe gets hotter because energy is dissipated in viscous flow. You could in principle calculate the energy dissipated using the pressure drop per length of pipe, which is described by the Darcy-Weisbach equation, but this would be a somewhat involved calculation for real pipes/taps and in any case it isn't relevant to the core of your question.
When you relate velocity to temperature you're presumably thinking of the Maxwell-Boltzmann distribution for the temperature dependance of the velocity profile in gases. The trouble is this distribution is arrived at by considering redistribution of energy between gas molecules due to collisions between them. If you simply add a constant velocity to every gas molecule you aren't making any difference to the way the gas molecules collide with each other, because it's only their relative velocities that matter.
Although water is a liquid not a gas the same argument applies. It's the velocities of the water molecule relative to each other that determine the temperature. So just adding a constant velocity to every water molecule makes no difference.
To greatly simplify John's answer: the temperature gained by friction and velocity is insignificant compared to the current temperature of the liquid.
I expect frictional heating would kick in at higher pressures, but once the water had enough kinetic energy to heat up noticeably on impact with your hands, it would also have enough energy to strip the flesh off your bones, cut through said bones, and punch a hole through the sink. Water jet cutters will easily go through inches of steel and a foot of stone - they aren't overly concerned about heating.