1) First off, if you're not exercising much, you should probably spend your willpower on ramping that up. In general, research has shown that willpower acts more like a muscle than a purely "fuel tank" model and taxing your willpower in small increments can actually strengthen it in the long run. [1] But in general, your shower time is already allocated for, and happens at a regular interval, so after exercise (which takes up time that people sometimes say "they don't have") it is a good place to do some willpower strengthening.
2) In the book, I actually explain that I start cold for 3 minutes and then end warm/hot. It's actually better than just going warm right away.
3) I admit that when the water is cold, you are not able to think about much. But if you turn it back to warm after a period, you still can spend the rest of the shower thinking. Also I find that clearing your head via the cold means the rest of your day's thinking is more clear, which may be worth the trade off.
Ultimately, you should just try it once, go through it, and see if it serves you.
[1] http://blog.bufferapp.com/what-the-research-on-habit-formati...