Why can't they accept 'the doctor has blood on their lip'. Sin means ownership and the gender was not specified.
Yes, but in English, "their" can be used for plural but also singular when gender is unspecified.
For the purposes of this course I'd just recommend choosing a gender in these situations. English pronouns are pretty complicated already.
If the object you mention belongs to the subject of the sentence (in this case to the doctor), you need to use the reflexive possessive sin/sit/sine. If you say "på hans læbe", you assume that the doctor has blood on someone else's lip, which would be a pretty weird sentence.
It's also pretty possible to just write "Lægen har blod på læben".