"Italians rarely take milk in their coffee"
Wrong. A "caffé macchiato" is pretty common, and you usually see a small pot with cold milk on bar counters, if you want to add yourself it (you can also ask for a "caffé macchiato caldo" - where hot milk is added like in a cappuccino).
Cappuccino itself is coffe with milk. "Marocchino" too has milk, adding cocoa also.
A "caffé corretto" (usually with brandy or grappa, but other liquors can be used) is usually an after dinner, but it's not uncommon to see people getting it in the morning, especially in some areas...