Does this count as a regular medical checkup?
No. Dr. Haugen is performing a regulatory exam as a Designee of the FAA to establish if you meet the standards for an FAA Medical Certificate. This is not a regular visit with your doctor, does not substitute for your regular doctor in any way, is not for the diagnosis or treatment of disease, and does not establish a doctor-patient relationship. The FAA is quite clear about this.
Does Dr. Haugen work for the FAA?
No. Dr. Haugen (and all Aviation Medical Examiners (AMEs)) are physicians who have been designated by the FAA--as specified in the Code of Federal Regulations (C.F.R.)--to perform services on behalf of the FAA.
Does the FAA set the fees for a flight physical?
No. FAA does not set or comment on fees that AMEs charge, other than that they should be resonable for the geography in which the AME is operating.
How can I speed up getting my medical certificate?
Visit the Read Me Info page and follow all of the relevant instructions to give yourself every chance of avoiding a deferral of your exam to the FAA. As you will read, though, there are numerous conditions for which Dr. Haugen is obligated to defer to the FAA, and by regulation, will result in him not being able to give you your certificate on-site.
If you need to supply any records, make sure you have all of them the first time, as every time the FAA needs to ask you for more info, it will add roughly 8 weeks to your wait time.