Why "that man over there" was not accepted? Is it wrong, since we don't have further context to distinguish between man and person?
I'd assume that we use person here, because we don't know their gender for sure.
So, without context, it should be translated as person, but with context can be translated as man. Is that it?
If context gives away the person's gender - yes. Like we would use "man" or "woman" in English. For example: вон тот человек - that person over there; вон та женщина - that woman over there; вон тот мужчина - that man over there
Why is it without 'to be'? I mean it looks like it should be "that person is over there"
That probably would be something like Тот человек там, or even Этот человек там. Вон тот appears to be an idiomatic phrase which means what the translation says it means.
"that", not "thot"
I'm getting the impression that you use тот in specific cases of pointing someone or something out who/which is not near you, but in idiomatic phrases like this one. Этот seems to function as both "this" or "that" is in a lot of circumstances, and that concept is reinforced by comments by some native-speakers.
Pretty much. When pointing at a distant object you can use вон тот but it often awkward to refer to it as simply тот. We only use "это" to present an object ("Это её ноутбук")—outside poetry, at least.
Тот is quite popular in grammatical uses to connect clauses. You cannot replace it with этот in this use. The structure is similar to the formal and old-fashioned English "that which".
- Это то, что я просил?. = Is this what I asked for?
- Скажи тем, кто не придёт, что они уволены. = Tell those who do not come that they are fired.
- То, о чём мы вчера говорили, крайне важно. = What we talked about yesterday is extremely important (lit. That about which we talked )
The same goes if you need то as an adjectival modifier when expressing "the one that...", though you may live without it (negation, though, makes it trickier):
- Это не тот художник, про которого я читал. = This is not the artist I read about.