Some verbs in Russian take the dative case where us non-native speakers might expect accusative -- my hunch is that it has to do with their intransitivity. Off the top of my head, I remember that Помогать behaves this way as well. There's a bit of information on the topic in the tips and notes of the "Dative and Plurals" lesson :-)
Yes, but it has a distinctly different meaning.
I believe him means that you trust that what he says is true.
I believe in him means that you trust that he exists.
"I believe in the Devil" is a theological claim that there is such a thing as a personification of evil, not that you trust what it says.
I believe in this girl is most likely to mean something like when the person you have just accused of theft claims that they are innocent, and a girl, whom you have not seen, took the item,. you believe that the accused person is telling the truth about the girl.
I'd consider "in" to be a particle here and the verb here to be "believe in" rather than just "believe." In this interpretation, "in" is a particle and "him" is a direct object. Phrasal verbs like "believe in" are common in English: lay out, take over, set up, break down, etc., and they are cousins to the (very interesting) separable verbs in German. See the brief article www.teachingenglish.org.uk/article/particle
I see where you're going with that, but it's the difference between "I believe her, she's telling the truth" (no "in", and Dative used in Russian) and "I believe in her, she has a strong character" (both Russian and English will have a preposition, and the Accusative is used in Russian).
They are two different things and only one meaning is correct here.
I think English is an exception among most languages in having accusative "believe" . Consider that "believe" can be applied to a story, a person telling the story; both use accusative in English. A deity triggers dative (He believes in God) as does having good thoughts about a person's future (I believe in you). It is quite difficult to use this verb in English.
Yes, because believing in someone means you have faith in them or a certain level of confidence in their ability, or even that you believe that they exist (for instance, "I believe in the flying spaghetti monster"). Simply "believing" someone means you think they are telling the truth about something.
Verbs that take dative include to help, to advise, to like ,some reflexive verbs , to need and verbs that represent the concept of sending or communicating something to ie to write to, to bring to , to call by telephone to, to tell her, to give to. Sorry I have no Russian keyboard on my computer.
It is the dative of эта. The dative of этот is этому. You can see declension tables in wiktionary, something that I strongly recommend:
Note that many other cases of эта are also этой, and этому is also the dative of neuter это (most, but not all, masculine and neuter forms tend to coincide).
In English, one could use this sentence to mean "do you believe that this girl is telling the truth?" or it could also be used colloquially to say: "check out this [crazy] girl. Can you believe that she's is [behaving] the way she is?" Might this sentence also have that same second interpretation in Russian?
Nope. Believe means "think [sth] is true/exists". Trust is more complicated. It means "believe [sb] doesn't/won't lie" or "expect safety/reliability"
I know you're not lying (in this particular situation), so I BELIEVE you. You have never lied to me or hurt me, so I TRUST you. (=I believe I am safe/I believe you won't lie)
The verb верить can govern two cases - Dative or Accusative. When Dative, it means "believe" in the sense of "trust" (like do you think she is telling the truth?). When Accusative (with preposition B), it means "believe" in two senses - either "believe in the existence of something" (верить в Бога), or "believe in someone's/something's ability" (--Я не могу плавать. Я боюсь, что я утону. --Ты можешь это сделать, я верю в тебя).