Do doctors get paid by insurance if the patient received services but didn't pay their deductible, co-insurance, etc?

At the beginning of the year, the insurance company will process the initial claim(s) and attribute the appropriate amount (covered services) toward the deductible. Until the deductible has been satisfied, the insurance company isn't going to pay a penny to either the patient or the doctor.

For example, traditional Medicare pays 80% of the bill. The doctor submits the form and Medicare remits the 80% payment. If the patient does not have a supplemental plan, he owes the MD 20%. If he has a supplemental plan, that insurance company remits the 20%. Private insurance is just a variation of this whether it be a percentage or a fixed copay amount.

The insurance company does not care about the copay. It is the doctor's responsibility to collect it since it is the balance of the money due him.

The documentation or proof of services, submitted by a provider to an insurer in order to receive payments, does not require that a patient has already paid what they owe. Patients typically receive their final bill after the insurer has paid its part. Patients might negotiate payment plans with the provider or might default.