Will cold water result in cooler coffee with drip coffeemaker? In electric drip coffeemakers, water is pushed up by boiling a portion of it at a time, right? 

My friend told me that unless he uses warm of hot water in the coffeemaker tank, the resulting coffee will be much cooler. I find this very hard to believe as I always use cold water and the coffee comes out hot. 

All the water will travel past a heating element, and steam "bubbles" will push the water on the top where it will "drip". BUT is it possible that in cheaper coffeemakers the cold water can slip past the heating element un-heated and travel up with the following bubble, cooling down the outcome? Is this where the watts will really come into play?

I am not asking if cold water should be used, but rather if such a scenario is possible. I do not have a precise thermometer or a cheap coffeemaker to test this. Also, I found this very hard question to google, as the only keywords I can think of are "cold, water, drip, coffee".
I hardly believe this is directly related to coffee. Still, as a more than 15 year experienced engineer, I would like to answer this question.

The answer is simply and clearly "no".

For the water to be able to travel to the top of the drip coffeemaker machine, it requires to gain some latent heat by changing its state to gas phase first. That's It simply means, it needs to be vaporized first.

Then, just at the top of the coffee and filter, there is a condensation space. At that space the gas (vapor) returns back to liquid phase (water) and starts to drip. Therefore, you obtain a slow flow of heated (to the near evaporation point) dripping water.

I cannot think any other way of bypassing this flow. Even in cheaper drip filter coffee makers.