1. "Can my employer require walk-in shifts that differ based on position and that do not apply to herself when she is working as an employee of her own company?"
Yes, your employer is allowed to treat different employees differently based on their jobs. She can even treat different employees in the same job differently as long as it is not based on an illegal motivator (like discrimination based on race, gender, religion, age or disability). She can certainly make special rules for herself as owner.
2. "can she require all commissioned employees to clean the spa every day when she will not."
Yes. She is allowed to treat herself differently, even if acting in a role as an employee.
3. "Is this gender bias or discrimination?"
It is gender discrimination. But in this case, it sounds like it is related to a rational business purpose, and as such, it probably is not illegal. If it is a fact that most clients prefer to work with female therapists, then she is allowed to discriminate in that practice. So it is technically gender discrimination, but if it serves a reasonable business purpose, then she is allowed to do it.
I know you want some references to tell her that she is acting
illegally, but she isn't. Sorry, I wish I could tell you otherwise... but I have a responsibility here to give the truth- the correct legal answer, rather than tell you what you hope to hear.