Translation:The majority of the people believe that I am crazy.
I didn't say you can't say it, I just said it doesn't sound as good in this sentence. Its use in the example you gave is because it's a cliche, and it also has parallelism with the "of the" in some of the time.
Again, it's perfectly acceptable to say "most of the people," but when you're using a computer program that has to have every one of the possible answers input by the staff, you're probably better off using the more common/closer translation.
I said that as well. Sometimes I smooth out the literal translations and it judges those wrong so I try to err on the literal side. I also try to imagine a situation where that particular literal variant would convey some subtle meaning, say you'd just walked off the stage after giving a presentation about a perpetual motion machine to an audience and your hermano asks you how it went. You would, in that case, answer "(sigh) Most of the people think I'm mad." When "the people" is a specific group in a specific place, not all people in general, the literal translation is more correct. It's certainly a more common phrase than "You can take the chain to the hotel," or even "bear aganst horse." ;-)
That's actually incorrect. In English, "majority" can be singular or plural depending on the context. In this case, since it is referring to a group of individuals, it is plural. A further explanation can be found at this site: http://public.wsu.edu/~brians/errors/majority.html
"People" is always plural, and "people thinks" is always wrong. But the subject of the sentence is not "people" but "majority", which usually takes a plural verb in English (unlike Spanish, apparently) if they are being considered as individuals, but can take a singular verb if they are considered as a single group. For example, if a political group were acting in concert, one might say, "The majority is voting in favour," but "are" would also be fine, and "The majority of the delegates are voting in favour" is to be preferred to the equivalent sentence with "is."
Those are technically the right definitions, but it's not really so black and white. Creer and pensar are actually interchangeable in most cases when using "think" to denote what your opinion is or what you believe to be the case (just as "believe" and "think" can sometimes be interchangeable in English as well). I think John is at home = Creo que John está en casa = Pienso que John está en casa. But I believe "creo que" is actually the more common construction here.
However, pensar is always used to denote reflecting or "thinking about something" as in Pienso en mi padre = I'm thinking about my father.
This is a tricky point. The word itself is singular but implies a plural. You will see it used both ways and I'd suppose they're both correct, although I typically use the plural, as I consider it in the same group as the word "people", a singular word that implies plural. "The people don't like it" uses the plural verbs and I think it sounds much better than "The people doesn't like it."
Be aware, however, that words such as this (e.g. gente) in Spanish will always use the singular verbs.