Does it matter if I start boiling hard eggs in hot or cold water?
It depends on how fussy you are about your hard boiled eggs. If you wait until your water is boiling and then add the eggs for the same amount of time, every time, then you will get consistent results. If you add the eggs to cold water a variety of circumstances (actual starting water temp, stove setting, size/style of the pan) will make your results 'inconsistent'. 

If you know you like a 12 minute hard boiled egg then you need to start with boiling water or you will get runnier eggs. Once the water reaches 212°F (100°C) it will remain there so long as the same amount of heat is being used. The boiling point of your water (based on other chemicals in the water and altitude) becomes a constant temperature to cook your eggs.

For 'consistency' yes, it matters.