Would 15mg zinc daily cause copper deficiency?

Depends: The National Institutes of Health considers 40 mg of zinc a day to be the upper limit dose for adults and 4 mg of zinc a day for infants under age 6 months, however, there are studies showing that long-term (>60 days) supplementation of zinc can lower your copper levels but that's probably at high doses. If your dietary zinc intake is <25 mg you should be fine.