lol the first thing I thought of was "I troll you".. obviously not correct ;)
That is also an accepted translation. You could maybe argue that I believe in you is more like Jag tror på dig, but we've chosen not to be picky in this case. I think there's a bit of overlap in how these expressions are used.
tror är antingen 'think' eller 'believe' på engelska. (ps. is that correct? = stämmer det?)
Yes, but then you need to include på: Jag tror på Gud - 'I believe in God'.
I'd say this is a pretty specific English expression, right? There are some Swedish idiomatic expressions that fit in that context, I'd choose either Vi sätter vår lit till Gud or Vi litar till Gud. The translation they give in Wikipedia is I Gud vår förtröstan which sounds archaic to me. None of those 3 are expressions I'd recommend to learners to use in other contexts though.
I would expect there is some relation between 'tror' and 'trust'. For example when one says: 'i trust you are telling me the truth'
Tror brings to my mind old fashioned stories where the characters say "I trow." Cognate word, perhaps?
While it obviously depends on context, in absolute general I would say yes since it can also mean to be under the impression that something is true, as in "Are they married?" "I believe so."
'you' is an object here so it must be in the object form. They just happen to be the same in English, but compare to how you would not say 'You believe in I' in English.
What's the phonetic difference between "dig" and "det"? I'd hate to say "Jag tror inte det!" and have someone think I was calling their trustworthiness into question.