It's explained elsewhere (under Tips on the mobile app, I think) that in Spanish, one would normally use el/la/los/las before parts of the body.
Possession of body parts is stated with an indirect object (se) and a determiner (las): "El niño no quiere lavarse las manos."
Interestingly modern English often uses 'their' instead of 'his' where the gender of the child is unknown. This was once considered wrong and 'his' was said to include male and female. These days gender sensitivity points to the use of a neutral choice, hence 'they' or 'their' depending on context. I didn't trust duo to be up-to-date so chose the translate niño as boy and avoided the issue!
It is ONE child, not more. Therefore his is correct. 'Their' would be correct with children.
No, it's now generally considered acceptable to use "their" to refer to an indefinite third person singular antecedent so "their" is also appropriate for "child", of which you don't know the gender. English lacks any other word, so unless you want to say "he or she" or rewrite the sentence to make the gender clear, you're stuck with "their".
I wasn't expecting to need to use the reflexive version of the verb here; lavarse instead of just lavar. I imagine I still would've been understood but any clarifying info would be appreciated.
You could use just lavar if he were washing his bicycle or his dog. But the se is used to indicate that it's himself he's washing, so wash his (own) hands needs lavarse.