Why can't we use 'nincs' here?
Because "nincs" means "there is no". So it would literally become "This is there is no dream".
But it seems that rarely is Hungarian translated literally but rather has many colloquialisms. I thought I was supposed to use "nincs" every time that I ran across "it is not."
nincs is more or less like nem van -- and so like van it is used in situations such as "there is [an object in a certain location]" but not in situations such as "X is (a) Y" (the cat is red, the book is a novel, this is a dream", etc.
Yes, it is "there is not", rather than "it is not". It is an existential thing. Something does not exist or is not located somewhere.
Yes, and also nincs is used in the 3rd person format :)
Would "nincs álom" be "there is no dream"?
Why can't we say "Nem ez álom?"
This would mean 'Not this dream' But instead it would be ' This is not a dream' Because 'nem ez alom' will make it sound like you're talking about a specific dream rather than saying it isn't (: