I don't have a coffee press so I just brew ground coffee as you would tea: in the cup.
How hot should be the water to brew with this method? Should it be boiling, or a lower temperature?
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Your method most closely resembles Turkish Coffee, which is suggested to be served at hot, but not boiling temperatures. Since the exposure to the grounds is so direct, it's easy to burn the coffee at boiling temperatures. You want a full body, but not a burnt taste (yuck!) Turkish Coffee World recommends about 158 degrees Fahrenheit or 70 degrees Celsius.
I'm no expert, but a frequent enjoyer, and I also use freshly ground coffee in a cup, then pour water over it. I always go with boiling water and have never heard of any reason to change. My coffee tastes great too, so I think I'm doing it right.
By the way, in my country (Sweden) we call that a 'Turkish' brew.