-12, -11, -10, -6, -5, -3, 3, 5, 6, 10, 11, and 12 are not
in the set S.  That is all that can be guaranteed.

Justification: Take the contrapositive of the two rules in
the problem statement.  Since 10 is not in S, 5 is not in S.
(Put another way: If 5 were in S, then by the 2nd rule, 10 would
be in S; but 10 is not in S, so 5 must not be, either.)
Since 12 is not in S, 6 is not in S, nor is 3.  Since these
numbers are not in S, neither are their negated versions.

Common errors:

1. "10, 20, 40, 80, 160, ... are not in S":
This is a converse error.  If the rule had said "If x is
not in S, then 2x is not in S" or "If 2x is in S, then x is
in S", this deduction would be correct, but that's not what
the rule said.

2. "S contains every integer except -12, -11, ..., 12":
Nothing about these rules forces S to contain any particular
integer.  For instance, the set {0,1,2} is a possible value of
S that's consistent with these 2 rules.