I tried to find out when to use of and about, but there was too little information to be sure so I want to ask you.
(meaning: What is your opinion of me?)
What do you think about/of me? (are both correct and mean the same?)
(meaning: You only care about yourself/your needs.)
You're only thinking of/about yourself/your needs. (are both correct and mean the same?)
(meaning: I was considering the problem, but I didn't come up with any solution.)
I was thinking of/about the problem, but I didn't think of any solution. (the first part: are both correct and mean the same? is only of correct?)
(meaning: I find you my friend.)
I think of you as my friend (is it possible to say "I think about you as a friend?")
I can't think of/about her name. (does it mean "I can't remind myself of her name."/"I don't remember her name." and work with of and about?)
(meaning: I remember the time we lived together)
I think of/about the time we lived together. (are both correct and mean the same?)
I'm still thinking about you. (is only about correct)
I've never thought of you (is only of correct?)
I'm thinking a lot of/about you. (are both correct and mean the same?)