Yes, if you want to peel them right away, you would cool to be able to handle without the discomfort of hot to the fingertips.I read that you put them in the cold water if you are peeling them right away. Is this true? Or not necessarily?
I want to peel one a day for 12 days.
The cooking time can vary by individuals doing it to their preference. Here is an OLE SKOOL timer.
When I used for 3 minutes, eggs came out SOFT BOILED. If I turned over for additional 3 minutes, they came out HARD BOILED.
I use a flame gas stove, so maybe using electric, and different amount of heat may vary your cooking time.

Experiment, and fine-tune your time and cooking temps.
I also encountered slightly green yolks when I cooked the eggs too long.
Here is a copy/paste from a website.
What causes hard boiled eggs to turn green?
The discoloration that sometimes forms around the yolk of hard-boiled eggs is the result of a similar reaction – this time between sulfur in the whites and iron in the yolks. It too, is harmless. It occurs when eggs have been cooked for too long or at too high a temperature.