1) more likely with "it", both possible.
2) requires "it", because "take it" means "understand", but "take" does not. Note that "understand that" or "understand it that" are both acceptable (probably parallel to (1), with the "it" being a little strange).
3) same as (1).
These seem colloquial to me, surprisingly: usually we get longer more syntactically complex sentences in writing. But for some reason these are casual rather than formal, although they are still part of a somewhat formal way of speaking. I just don't expect to see them in writing at all.
One analysis here is actually pretty simple: it's an appositive use of the CP. "I know that guy, John!" or "This is my friend Bill". It's just that in some cases "it" has become a fairly normal part of the sentence. We do get "expletive" it elsewhere in English, though, so at least some effect of that is probably relevant to why we get "it" here.
Anyway, if you remove "it" those sentences are fine, and most are, except in the case of "take it" because that is a specific phrase. But "believe it" and "know it" aren't.
By the way, note that in Spanish (only other example I can think of) you often get what appears to be an article on 'headless' relative clauses; this could be translated as an 'it':
Lo que quiero es una hamberguesa.
it that/what I.want is a hamburger.
'What I want is a hamburger.'
As far as I know (at least in some cases) just "que" without "lo" is ungrammatical when it's this kind of clause (object clause?).
