These are basically fixed constructions in English. You are friends with someone. Note the -s! Also, the with is not optional.
You are free to use whom or leave it out. Who is technically not correct in this case. Not everybody will consider it incorrect if you use who, though.
These sentences are all correct, read them carefully to understand how they are different and how they are similar:
These are people who are my friends.
These are people I am friends with.
These are people with whom I am friends.
People I am friends with, should be...
People with whom I am friends should be...
People whom I am friends with should be...
As for you other sentences, the verbs date and move in are not used in a similar way, so you cannot use them in a similar sentence! You date someone (not with someone), but you move in with someone (moving in someone would be possible, but it is a very graphical description that you might want to keep to yourself unless you are a baby moving inside your mother!)
She is not the kind of girl I would date.
She is not the kind of girl I would move in with.
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.
Best Answer
You are right; it is not necessary. However, it's not grammatically incorrect.
Most of the time, I wouldn't use at when expressing my age. However I can imagine myself in an occasional dialogue that goes something like this:
In this context, the person is giving their age as though they are telling where they are in long race:
It's not a common usage, but it's not an incorrect one, either. Still, you are correct to say that the word at is not needed. Most of the time, I'd recommend omitting it.