Several underlying issues must first be dealt with, before the OP's main question can be adequately addressed. For that purpose, I would like to start from the bottom up.
Is there a way to clarify this so there's no room for misinterpretation?
Yes, there is.
We limit the number of possible interpretations of a piece of discourse by providing context. Context gives us additional information, so that we may interpret a word, phrase, sentence, etc. with greater accuracy.
If you were to tell a friend of yours about how hot the steam is in your local sauna, chances are that he won't mistake the rising of vapor for the digital game distribution platform Steam. He will be able to arrive at the correct interpretation, because you've already provided him with context.
Next, let us look at the "Is it a boy or a girl?" example.
The above question is what is commonly referred to as a binary question. This means that the question has two* possible answers. And while some binary questions do call for "yes" or "no" answers, this example is not one of them. In the absence of any other contextual information, we can safely assume that the person asking this particular question expects us to answer with something along the lines of "It's a boy" or "It's a girl." A single "yes" or "no" will not answer the question in a satisfactory manner. Instead, it will only create ambiguity, leaving the questioner wondering about what you wanted to say.
*In this thread, a contributor explains in more detail how some seemingly binary questions often have a third answer - what they call the "It depends" option.
If someone asked me "do you think that people are dumb?" and I said, "no," I mean that I don't care about it - not that I think the opposite: people are not dumb.
The reason you feel that your conversation partners are misinterpreting your answers, as in the example above, is because you are attempting to go for the "it depends" option, without providing context for the questioner. Unless your interlocutors are able to read your mind, they will take your "no" answer at face value, or in other words - people are not dumb.
To safeguard yourself against ambiguity, try to define your answers more clearly to the questioner.
Q: Do you think people are dumb?
A: No. / Yes.
Q: Do you think people are dumb?
A: No, I just don't care about these kinds of questions.
I understand "I don't want to be happy" as that you don't CARE about being happy - not that you want to be unhappy.
Here, you are already on the right track to answering your own question, Sam. Compare the following:
2 a : to have a strong desire for
e.g. wanted a chance to rest
1 : to be concerned about or to the extent of
e.g. don't care what they say
e.g. doesn't care a damn
If the person wants to state that they are not concerned about being happy, they would use "care" to convey their desired meaning.
Likewise, if the person does not have the desire (the need) to be happy, then "want" would be the more appropriate word.
Do "I want to be unhappy." and "I don't want to be happy." mean the
Short answer: No.
Long answer: This, again, will depend on the context.
To make things even across the board, let us set the following question as our baseline:
"What do you want to be?"
In "I want to be unhappy.", the speaker's focus, their end goal, as far as we are able to interpret from our contextual baseline, is to achieve a state of unhappiness.
Should the answer be "I don't want to be happy.", we may interpret the goal of the speaker to be a great number of things. However, one thing we can be sure of, again, given the context we've set, is that he absolutely does not wish to be happy.
If there is one takeaway point for you from all of the above, it would be this: context is key.