I'd say of course De trodde mig inte -> They did not believe me and De trodde inte på mig -> They did not believe in me are better translations, but the difference is small and we accept or should accept both ways on both sides.
You can actually start a sentence with De trodde inte mig …, but only if you mean They did not believe me (… but they did believe someone else). That one is not accepted here and I don't think it should be either, since that sentence doesn't really work on its own in Swedish – without a continuation it just doesn't sound right.
I have a question. When do I use 'inte' after the words like 'mig' and 'dig'? I saw some sentences that use 'inte' right after the verb but others use it after words like 'mig'. Could you help me?
P.s. English is my second language, so... I'm not good at English grammar... Sorry for saying "words like 'mig' and 'dig' "