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’).