Well, your assertions are not entirely correct. For example it's fine to say
The doctor asked the history of my illness
Use "about" when you want to know the general facts around some information, or when the direct and indirect object of the verb might not be clear. Example:
He asked the computer.
He asked about the computer
In the first example, the computer is the indirect object, from which he expects an answer. In the second it's the direct object, the subject of his query.
In the same way, it would be wrong to say something like, "He asked my mother's health," because that makes no sense. You can't ask anything of someone's health. Instead say:
He was always very friendly and often asked about my mother's health
Because adding "about" can imply that the information is doubtful or unusual, don't use it when requesting specific, well-defined information such as, "The doctor asked my date of birth." Instead use it when you want to know more peripheral detail:
The doctor asked about my date of birth -- whether there was an official record, and if I was sure it was the correct date. I think he was trying to say I might be younger than I think I am.
I asked about her name, and what it means in her own language, but she told me it actually doesn't mean anything. Her parents just liked the sound of it.
In some situations, however, "about" implies courtesy, as indirect questions may be more polite in certain contexts.
The customer asked the price of the car.
The customer asked about the price of the car.
Note there are many other ways to express this kind of polite deference:
The hesitant young man inquired into the price of the ring, and whether he might possibly pay in installments.