Does reduction from an NP-complete problem to some problem $X$ imply that $X\in NP$?
the reduction only shows that X is at least as hard as SAT (NP-HARD). This could mean that X is in a class suspected to be harder than NP such as PSPACE or EXPTIME. To show that X is NP-complete you must show that it is also in NP. If you try it out you will see that you can easily reduce SAT to problems in EXPTIME that are not suspected to be in NP. The idea being that your reduction shows that X is at least as hard as SAT, but there are classes of problems harder than NP problems which X could fall into.