Modern linguistics have shown that 'whom' is mostly disappearing from spoken language. 'Who' is considered correct even where in the past you would have had to use 'whom'.
Yes, but these days most people have dropped "whom" from their vocabulary and just use "who."
The "who"→"whom" part notwithstanding, that would be "Кому она доверяет". The meanings are slightly different in both languages.
You cannot say "Кто она верит?" for a very simple reason: Кто is nominative, and hence can only be used to question the subject of a sentence. But then "она" is also nominative, and hence it is the actual subject, making any questions with "кто" meaningless regardless of the verb used. What probably trips you here is the English use of "who", but it might be helpful to remember that in better English the question should actually be "Whom does she believe?".
As for choosing the case (once we have established that it cannot be nominative) -- that is determined by a particular verb, and yes, for "верить" it should be dative.
English uses the accusative case "believe whom." It would be the locative or dative case to say "to whom."
One could say "In whom does she believe?" It's perfectly natural to hear "I believe in...X" and X be a person. It just might sound a little melodramatic.
I believe (no pun intended) that "beliving in somebody/something" is quite different from "believing somebody". I don't believe in Martians, but I can't tell you whether I would believe them until I talk to one. Perhaps one could use "to believe in Jesus Christ" to mean "to entrust oneself to Jesus Christ", but such a use strikes me as highly dated/poetic, something along the lines of "in God we trust".
Oh you're absolutely right that it sounds poetic and dated. But I can recall TV shows for children where characters would tell others that they "believe in" them in order to show encouragement. Rare, yes, but it IS used.
Verb governance and use of prepositions in Russian is extremely difficult for some, such as myself. Thank you very much!