The most dangerous component of exhaust, the carbon monoxide that everyone has mentioned, has no smell. And that's the one that can cause all of your symptoms.
IMHO a smoke test is inadequate to test for a problem of this nature. The system needs to be at operating temperature and under normal exhaust pressures. IMHO a carbon monoxide detector or a combustion analyzer (which also measures carbon monoxide) is needed here.
But I sure wish you'd answer some of the questions. Like are you the new owner of this, a used car? There are things people do in their cars, like smoking and hanging scented items on the mirror, that can cause serious allergic reactions in other people.