"They did not believe me."
Translation:De trodde mig inte.
When it's a pronoun, the normal word order in cases like this is to have the pronoun first and inte last, but both word orders are possible (De trodde inte mig is possible, but it stresses me in a way that implies you're contrasting this to something else you're going to say, that they believed someone else instead). With names or other nouns, inte must go before the noun/name. This is how it works generally, in sentences like
De förstod mig inte or De förstod inte mig
but De förstod inte barnet (impossible to put inte last)
In this specific sentence, I would actually avoid saying this sentence at all with a name/noun in it, in that case I would use the construction with tro på instead. De trodde inte på barnet 'They did not believe the child'.
Yes, it's only after -a that we tend to skip that ending. pratade is normally pronounced 'prata' in casual speech (it isn't wrong to say the ending either) but trodde is always said trodde.
It isn't. You can write either De trodde mig inte or De trodde inte på mig. Just like in English, They didn't believe me or They didn't believe in me. Both are correct.
It's pretty parallel so if you want to be accurate, you should translate believe in as trodde på. This one can be used for believing in someone's potential, which the one without på/in can't.