In countries with universal health care, can a doctor open a private practice?

Yes,

The illusion of a better quality of health care through paying out of pocket is just that, an illusion.

Here in the states, we pay out of pocket and our healthcare is ranked somewhere like 63rd or 64th in the world. The top 10 countries on that list are all under a Universal system.

Yes. A doctor is generally free to open his own business and not subscribe to the universal health care system in operation in his country.

Countries that have universal health care also have a private care system that runs much like any insurance based, service on demand system.

In Canada its actually against the law to open a private for fee practice unless the procedure you're performing is not covered under the universal health care system.

Examples would be dentistry, optometry, and non-essential plastic surgery.

Yes. I know that in Australia, which has a universal medicare system, doctors do have private practices. In fact my cousin's wife is a doctor in Australia and she is in private practice.

Yes, a lot of doctors on the NHS in the UK do private work. That's likely why the waiting lists are so long!