can't the translation be: you aren't a man?
No. We can accept a declarative word order for a question, but not when the Polish question had "czy". If there's "czy", then it really is a yes/no question and not something like "What?! You aren't a man?!"
Why do I need to use "are" twice in this sentence? Why can't the sentence be "Ty nie jestes mezczyna?" or "Czy ty nie mezczyna?"
"Czy" is optional. "Czy ty nie mężczyzna" doesn't sound good, use "to be" verb.
"Czy" isn't actually a verb. It's an conjuction, used for example (optionally) at the start of the yes/no question. Later exercises should also show you more of its uses.
Can one just skip 'czy'?
Usually yes, although dropping it in this "negative" question would probably give it more of a surprised tone.
"Czy ty nie mężczyzna?" is like saying in English "You not a man?
I assume they use this phrase to trick men into doing something in Poland too?