Does a doctor have the right to charge you more than the copay on your insurance?

That depends on 2 factors..1. is your doctor in your insurance company's network if no then yes he/she can charge you up to the billed charges subtracting what if anything your insurance company paid.

2. Is things like labs, x-rays and procedures covered under your copay or do they apply to your deductibles and coinsurance?

When in doubt contact your insurance company