Do I Have Exophthalmos? (photo)

Yes you do have relative exophthalmos, which in your case is inherited.  You may benefit from (cosmetic) orbital decompression to settle the eyes back.  See my website link for more information.

Your cheek bone is relatively weak making the eyes look more prominent.  Treatment options range from advancing the mid face (craniofacial surgery) to orbital rim and cheek implants.  Non surgical treatments can be remarkably effective and this is performed with hyaluronic acid fillers to augment the mid face area.  You appear to have approximately a 2.5-3.5 milliliter volume deficit on each side.

Based on your photos, you do have prominent eyes in relation to the cheek/midface region. This is not an uncommon anatomic variation that some people have. If you feel that your eyes have changed in their prominence, then thyroid eye disease is a possibility. The normal thyroid hormone levels are reassuring, but do not completely rule out thyroid eye disease:  about 5% of patients of thyroid eye disease have normal hormone levels.

However, the most likely scenario is that you have "shallow orbits". That is the bony volume of your eye sockets is small in relation to the soft tissue volume [eye/muscles/fat]. Unless you are having problems with dry eye due to your inability to fully close your eyelids, most surgeons will shy away from "orbital decompression" surgery in asymptomatic patients.

However, orbital decompression has been done for elective, cosmetic purposes as well. Make sure you pick an experienced oculoplastics/orbital surgeon.