It’s great that you want to get your zones dialed in. It’s by far the best way to keep track of how your body responds to training, and adding in your cycle can be another puzzle piece to keep in mind as you plan. The first week of the cycle (day one being the first day of your period) are when hormones are lowest, but that can also be a time that some women don’t feel great. Estrogen peaks right before ovulation (considered the 14th day of a “normal” 28 day cycle), then drops as progesterone starts increasing until the end of the cycle. What I’ve seen with athletes I coach and my own cycle is that the first almost two weeks are when energy levels are highest, and it’s easier to do hard workouts (including strength training). Aerobic runs feel good and easy and we feel the best. Then ovulation happens, there’s a slight drop in energy, then another week of feeling relatively good, followed by a week not as great as progesterone increases. So to answer your question, if it works, I would do the test in the first part of the cycle. In terms of the heart rate being affected, our body temperature rises at ovulation, so that could play a role if it’s a hot day, by increasing the temperature further. I personally have a very low temperature, so I’m not affected by that so much. Obviously there is huge variation and so take all of this with a grain of salt, and the best way to find out how you respond is to keep track of it with your training log. If you use Training Peaks you can add Metrics to the days that you feel certain ways, or when you start your cycle, and start noticing patterns.
Since the aerobic threshold test isn’t an all out effort, you can retest it often. The HR number will vary day to day for everyone, so it’s good to have a sense of how it should feel as well as what your monitor says. It might be helpful to do tests at different parts of your cycle to see how it changes. As with so many things, there’s no set answer, but I hope that helps!
PS-to answer your fasted or fueled question, I would go with what you normally do to get a sense of what that is. In general I don’t think fasted workouts are as effective for women as men.