Yep. Although 'trust' is more commonly translated «доверя́ю», and Duolingo might not allow 'trust' as translation for «ве́рить» so that people learn the main meaning of these words, and not the secondary meanings. Or it might allow it. No idea (I don't use the actual course, just going through the questions :D).
Никому is the dative form of никто, никто is the nominative form. You can’t use nominative form here (unless you want to say that ‘nobody believes’).
It is a double negative. We got told somewhere early in the course that Russian pretty much always uses these double negatives where in English we would use a single negative (and a double negative would be wrong). Just have to get used to accentuating the negative when thinking like a Russian!