For the meaning of "wanting to know someone's thoughts about something" the only good sentence is (6).
Let's look at the ungrammatical ones
(1) I asked him on an opinion.
(3) I asked him of an opinion.
(7) I asked him over an opinion.
Each of these sentences is ungrammatical. The prepositions used don't work in context.
(5) I asked him an opinion.
This is also unacceptable. In this construction, the sentence has the meaning of I asked an opinion to him. Now, you can't ask an opinion, it doesn't make sense. "An opinion" becomes the direct object...and opinion doesn't make sense as the object of "ask".
Now let's look at some of the correct, but odd sentences.
(2) I asked him about an opinion
This is a correct sentence, but it doesn't quite mean what you think. In this sentence, it's assumed you already know the opinion, and you are looking for more information surrounding it—for example, the reasons for it.
(4) I asked an opinion of him.
This is an acceptable sentence, but again, the meaning is slightly different than what you give. It doesn't mean so much that you required him to give his opinion, but rather that you expected him to.
The two definitely have overlapping meanings, and I think which is more common may depend on dialect. To me, as an educated British person who's been exposed to a lot of American culture (haven't we all?), the difference in meaning is very subtle, and reflects some psychology.
"To ask someone on a date", I would expect to see where the person doing the asking has some relatively specific idea. It suggests confidence, though not necessarily excessive confidence. "To ask someone for a date" suggests the attitude of a supplicant, as if the other person has something you want, and you hope they deign to give it to you.
(Note that either may have different meanings outside the context of 'dating', i.e. romance; "to ask someone for a date" particularly can have uses other than this.)
Best Answer
Asking about something refers to a topic, for example, you would ask a coworker about their weekend. You can't actually get someone's weekend activities, as it is not an object, but you can learn more about what they did, where they went, etc.
Asking for something refers to an actual object, for example, you would ask for a hamburger at a fast food restaurant. You can ask about a hamburger, but you would be asking about the details of it, like the ingredients, if it is organic, etc.
For your specific sentence, I would use for instead of about. You are asking for an object that you will be using, not details about the number.