I really disagree with the choices on this question...

Question

The question was «Oui, je le crois» CANNOT mean...
Yes, I think soYes, I believe soYes, I believe himYes, I believe <- the given answer

I can't speak for how English is used everywhere in the world, but at least where I'm standing, there is absolutely no difference between "I believe", "I believe so", and "I think so." They mean exactly the same thing. The choice that is the most different (by a tiny margin) is "I believe him."
I'm guessing that the idea you were going for is that if I'm believing "him", I'm not believing "in him", but I'm believing an idea that he has previously presented... but you didn't show any examples of that type of usage in the lesson. So given my choices, it looks like "le" is replacing a person in that choice.
My other questioin, of course, was a very similar question with similarly confusing answers.
«Il ne le croit pas» CANNOT mean...
There are often times on this site when, as far as I'm concerned, two choices are equally correct in English, but I can usually tell from the context which one you WANT me to pick, based on what the topic is. But these particular questions are a bit odd to me because leaving off the word "so" makes no difference at all.

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Sorry for the messy formatting. Apparently the script decided to strip out my carraige returns when I copied and pasted.  :P

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Hi Donovan - thanks for this feedback. I've taken a look and I agree these questions are more confusing than helpful. I'll flag them so they don't appear again until we can look at how we could rephrase these to make the intention clearer or find a way to better test that students have understood the underlying principle.

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There is a difference between croire and penser in French though. You are dealing with "un degré d'opinion". On the scale of things trouver is beneath penser and croire is stronger than both of them. So while they are pretty much synonymous in English, "Oui, je le crois" cannot mean "I think so" even though that distinction doesn't exist in English.

"Je le crois" being "I believe HIM" was taught in another lesson, so including that in these questions and assuming that this is a lesson you have covered without liking it here was probably a bad choice.

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Repeat post, sorry.