(1) Is your name, copycat?
That is not correct. The comma sets off "copycat" as a "noun of direct address". In such cases it should be possible to omit this word and what's left should be a complete sentence. But "Is your name?" is not a complete sentence. Is WHAT your name?
If your intent is to ask whether the person's name is "copycat", you could write
Is your name copycat?
If the idea is that some other name has been mentioned, then you need the word "that" to refer to the other name.
mumble mumble Bob mumble mumble
Bob? Is that your name, copycat?
(2) There is an animal that lives by night.
The point of this sentence is to claim the existence of an animal that meets this condition. So the "base" of the sentence is "There is an animal." We then modify animal with the phrase "that lives by night". This phrase is functioning as an adjective. When we do this, we must introduce the phase with a "refer back" word like "that" or "which". Otherwise the sentence could become unclear, whether we are saying that something exists -- "there is" -- or describing the actions of the animal -- "lives". The sentence would have two predicates. I suppose in this example it would still be clear what you meant. But in other sentences it would not be. For example:
Bob loves Sally lives in Boston.
Who is it who lives in Boston, Bob or Sally? Does Bob love Sally, or does he love the fact that Sally lives in Boston? We need some extra words to clarify which is meant. Like, "Bob loves Sally who lives in Boston", or "Bob loves Sally and he lives in Boston" or "Bob loves the fact that Sally lives in Boston".
(3) He was saying that your ideas are false.
This is similar to (2). Without the "that", you have two predicates. Is he saying, quote, your ideas are false? Or is it false that he is saying your ideas? In practice people often omit the "that" in sentences involving "saying" and similar words because it is understood that the action is "saying" and the remainder of the sentence is essentially a quote. So you could get away with either form in (3).