When is ...だと思う and when ...と思う used? Do I understand correctly that the former is used after a noun and na-adjactive only, and the later after a verb and i-adjective?
I like this question a lot, and I think we can answer it with a better understanding of how the と particle works.
In this case the「と」particle is actually a quotation particle. The formation for use of this particle is relatively easy:
(quoted sentence with plain form ending) と (Your verb. i.e. 言う、思う、聞く、etc)
So really what comes before the end of the と particle in this case is the plain form verb. I think that several examples will help you out.
彼女｛かのじょ｝は綺麗｛きれい｝です。(She is pretty)
綺麗｛きれい｝is a な adjective. Using it at the end of a sentence, we get です, or in plain form だ instead of な。
彼女は綺麗だと思います。(I think that she is pretty.)
Taking the first example sentence, and putting it into plain form, we can add と思う。 Without any complications.
あのビルは大きいです。(That building is big.)
大きい is, of course, an い adjective. In the sentence I ended it with です (to be polite), but です in this case is unnecessary. I could say
あのビルは大きい and still have it mean the same thing. Plain form with い adjectives actually end with the adjective, which in this case is 大きい。
あのビルは大きいと思います。 (I think that building is big.)
I'll give you a few more examples without commentary, to hopefully solidify the point.
あの人は大学生です。(That person is a college student)
あの人は大学生だと思います。(I think that person is a college student.)
彼は十キロ走っています。(He is running 10 kilometers.)
彼は十キロ走ってると思います。(I think that he is running 10 kilometers.)