I still don't understand how 어떻게 can be translated to both "how" and "what" sometimes. So i can understand why it's hard for you to understand the differences between "how" and "what" in some situations.. But anyway :D
1: Yes. If you say "what do you think?" it could mean that you are asking for an opinion or it could be an answer to someones question.
A: "What is this?"
B: "What do you think?"
2/3: Same here but you don't use this when asking for an opinion. If you just say "How do you think?" it's usually an answer to someones question.
A: "How does this work?"
B: "How do you think?" - When you answer like this it could mean that the answer is obvious, or it could mean that you want the person to figure it out for himself.
But you can say, for example, both "how do you think of me?" and "what do you think of me?". There is a slight difference though. When you say "how do you think of me?" i would say you are asking about how that person feels about you. It would be like asking, "in what way do you think of me?". Possible answers would be:
"Like a sister"
"I think of you as a friend"
"i think of you only as a coworker"
"What do you think of me?" i would say is more about asking about your qualities or characteristics.
"I think you are cute".
"i think you are fat".
"you are a hard working person".
It's actually much harder to explain than i though.