I keep smelling exhaust fume-like smell everywhere I go. Should I be worried?

Yes. If you are the only one who is smelling exhaust fumes, then this might be a sign of a chronic nasal / sinus infection. Rarely, this could be caused by and adverse effect of a medication, migraine headaches that occur without pain, or even more rarely, a stroke. See your doctor to evaluate these possibilities.
Maybe. See your doctor. Have him look in your nose. And consider a ct scan of your head, .