Both "nic" and "niczego" are valid Genitive forms, so both "Nie wiesz nic" and "Niczego nie wiesz" are correct.
As you're addressing Jon Snow by saying that, you need Vocative. Now, Vocative is kinda vanishing with first names, so if it was just "You know nothing, Jon" then "Niczego nie wiesz, Jon" would probably be more common than the Vocative "Niczego nie wiesz, Jonie". However, here you have both the first name and the last name, and then you definitely do need Vocative for the first name: "Niczego nie wiesz, Jonie Snow".
For surnames, you do not use Vocative, and with the surname Snow (pronounced the English way of course) it would sound very weird anyway... "Snole"? I'm not even sure.
"I don't see anything" should be ok. Here common speach always uses contractions, sounds rather strange when you don't!
This is accepted. Such contractions should be accepted automatically by means of coding.
Is this more understood as "I cannot see anything" or "i dont see anything"? The recommended answer suggests the former.
Literally "I do not see", but you wouldn't really translate "I cannot see" any other way.