[Ordnael92]

it turns out they will ship the Zara package to my apartment will DNA still work if they knock on the door and I open and receive the package?

Yes.

Simply fake sign It with a name that's somewhat legible. As such, It will not be associated to yours when cross-checked, hence "you" didn't sign the delivery.

how long should I wait before contacting customer support?

If you meant "after receiving the package", just contact them the next day asking when you're going to receive it