It is possible this employee has misophonia. In which case he should get headphones. If you are doing a lot of running around at work, it might be annoying other people too- so just be aware of it. In the same way anyone would be aware of the noises they make at work so as not to bother colleagues.
I personally get annoyed at the sound of my own shoes sometimes in a quiet space. So I get it.
Co-workers can ask whatever they like. But it's illegal for a superior to penalize you in any way for not confirming to their own gender ideals whether that means wearing heels or not wearing heels regardless of your actual or perceived gender. The Supreme Court ruled on this decisively like two decades ago.