For the most part, you should never consider
separating layers stuck to your cel i.e.
background, foreground, sketches etc.  You may end
up damaging your cel or whatever you are trying to
remove.  If its absolutely necessary to separate
the layers, take it to a professional (frame/art
shop) and have it done.  As for the background
itself, its usually easy to tell whether or not
its original.  The paper will be a heavier stock
(copy backgrounds usually come on copy/printer
paper) and the colors should be vivid and fluid
and not dull or grainy.  Backgrounds can be
valuable but those are generally sold separately.
 Common things like moonscapes, forests,
cityscapes are popular because they're fairly
common in anime series, but most cels do not come
with them due to the ratio of cels to backgrounds
for a given scene.  Also, ask the previous owner.
 Although they may not know, it can't hurt to ask.
 Welcome to the hobby and happy hunting!