In the sentence "people like us never wash […] hands", should the pronoun be "our" or "their"?
Either works. Grammar is correct. Meaning is slightly different.
People like us never wash their hands
The speaker first includes himself by using 'us' then stands outside the group to use 'their'. This is grammatically fine, if logically dubious. The speaker is putting a little distance between himself and these unclean people at the end.
People like us never wash our hands
Still grammatically fine. Here the speaker adds no distance. The feel is almost conspiratorial. I almost expect an evil laugh at the end.