Nationality or Gender of Instructor
The race or gender of the instructor is completely unimportant. Do not assume a Korean Taekwondo instructor is better than any other instructor. Do not assume a male instructor is better than a female or vice versa. What matters is whether the instructor can instruct and inspire you to be the best you can be at Taekwondo
Aesthetics of Dojang
You do not want to train in a dump but otherwise the appearance of the dojang is not that important. Many excellent instructors are just starting their businesses and cannot afford the best location. Many times, being a part of a growing dojang and helping it expand is as rewarding as the training itself.
A good dojang needs a clean, safe training area, punch/kick targets, punch/kick bags, maybe some mirrors, and maybe some mats. Do not be overly influenced by unrelated stuff, such exercise machines, weight room, hot tubs, tanning beds, etc. If you want to pay for all these extras, that is okay, but they are unnecessary.
Do not be influenced by flashy uniforms. Be suspicious of dojangs that use uniforms that are too different from the norm.
Appearance of Instructor
A skilled Taekwondo fighter will probably be in excellent physical condition. A skilled Taekwondo instructor may be a physical wreck. When you are in need of medical attention, you do not care whether the doctor helping you is fat or not, you only care whether he or she can help you. Similar to good coaches in other sports, good Taekwondo instructors come in all sizes, shapes, and levels of fitness.
Trophies are nice for the people who earn them, but do not be too impressed by a large collection of trophies in a dojang. Anyone who goes to a lot of tournaments will have a trophy, sometimes you may be the only person there in a belt/weight/age class so you get the first place trophy. If you are interested in the trophy collection, read what is on each trophy to see is it is an important one. Also, consider the ratio of trophies for sparring and patterns. This will give you some indication of which the school gives the most emphasis, sparring or patterns.
There is no universal martial art ranking system. There is no one international governing body that uniformly and fairly assigns ranks. Unless you are familiar with the way Taekwondo rank is awarded in the United States and with the ranking system within the organization with which the instructor is certified, then do not place much emphasis on the rank of the instructor. Without any training at all, you can buy a black belt, declare that you have just created a new martial art style, and promote yourself to 10th degree black belt without breaking any laws. Be suspicious of those claiming unusually high ranks. Each major Taekwondo organization has its own rank structure and is usually in a power struggle with the other organizations, so it is difficult to compare ranks between them. So do not base your decision on which dojang to attend on the rank of the instructor.