Yes it can bc of certain things in cold meds as codeine that are given by the doctor. It has happened to me recently but the doctors giving the test will understand. Just make sure you didn't do any other drugs or meds especially illegal ones. However, if you have been clean from illegal drugs (meaning you have not smoked them since the las time you did), you're fine, and the only drug you have been taking lately is cold medicine, then cold medicines will show up.
Yes any of them, even over the counter allergy medicines. So can headache medicines like ibuprofen or naproxen sodium (Advil and Alleve, respectively). The drug-testing jobs don't really care about YOU getting the job and proving you aren't a junkie and aren't on anything illegal, they care only about weeding out lots of applicants and this is like the final "cut."
Certain types can, the ones you get from the doctor that have codeine in them. You could probably call your local pharmacist to find out.