Can you really use this phrasing to say brushing teeth in French? My intuition is that the phrasing is as weird as saying you clean your teeth in English.
Yes, you can say "se laver les dents" or "se brosser les dents".
you don't say "cleaning your teeth"? Go and clean your teeth! Filthy child. Sounds ok to me...
It's ok, but is not accepted as a translation.
I don't think that's what Awesomo was complaining about whether it was accepted or not though. I thought they were saying that "clean your teeth" sounds odd in english. I don't think it does though.
"This man is cleaning his teeth." is accepted as a translation.
Can you say something like: Cet homme lave sa dents?
No, you have to say:
FYI, there is a mistake in your suggestion: since "dents" is plural, it would be "Cet homme lave ses dents.", but it does not sound natural in French.
se laver is a reflexive verb, so "cet homme se lave les dents" already implies that he his doing it to himself. Therefore, it does not make sense to put "ses" there.
That was indeed the other thing which confused me here - so I assumed it had to be about false teeth and he was washing his own (with 'ses' added for emphasis) and could later go on to wash those belonging to others.