Once the out-of-pocket maximum is met, do I still have to pay for co-pays for regular doctor appointments?
You should read your plan description to make sure, but in all cases I've seen, no, you don't pay co-pays after paying your out-of-pocket maximum.
I have heard that some clinics will still require a co-pay if they can't verify that you've hit your out-of-pocket max, so I suppose in that case you'll get reimbursed by your insurance. Fortunately I've never hit my out-of-pocket max to experience that firsthand.
The current state of the "out of pocket maximum" actually indicating an insured's total out of pocket limit for covered services is a result of the ACA. Previously, "out of pocket maximum" was mostly a euphemism for "coinsurance maximum." In these pre-ACA plans it was very common that insureds would continue to owe copays to facilities, doctors, prescription drugs, etc. Under the grandfathering rules, the ACA allows pre-ACA plans to be retained by those who held them at the time the law was signed. The full umbrella "out of pocket maximum" is one of the very few provisions that carriers did not have to homogenize under the grandfathering rules, 3rd party appeals is the other big difference.
It's possible that you have a grandfathered plan and your "out of pocket maximum" is really a "coinsurance maximum."
Otherwise, as DStanley points out, this could be a simple time delay and you can request reimbursement.