Does NP-Complete imply non-satisfiability?
You're confusing languages and instances of them. A 3SAT formula could be either satisfiable or unsatisfiable. The 3SAT problem is to decide whether a given 3SAT formula is satisfiable. The 3SAT language is the set of all encodings of satisfiable 3SAT formulas. Only the 3SAT language is NP-complete. NP-completeness is a property of languages, not of their instances. If every 3SAT formula were satisfiable (or every one were unsatisfiable) then 3SAT (the language) would be very easy.