They do not.
All their teeth were pulled out when they underwent Spartan conversion. Thus, they are fed protein-rich gelatin through a tube.
All their hair has been shaved off, so there is none to clean.
Ears don't need cleaning because their bionic aural implants plug up the entire cavity.
They don't blow their noses, they were removed as well. Why would you need a nose when you live inside of a metal suit? The BO would get overpowering quickly.
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Their bodies are covered in and kept clean by the skin tight undersuit under their armor. There would be no NO. Hair grows back. Brushing your teeth means cleaning your mouth in general, not just your teeth. I think your making stuff up. Link please?
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Yeah, I'm BSing. Though they could have had the follicles removed.