Why is ham used here, not han?
he = han
him = ham/han
Both 'han' and 'ham' are correct. The course is trying to be consistent with the translations, so 'him' is always translated to 'ham' in the suggested solution. (Which is not better nor worse than 'han')
Thanks that makes sense now
Is it bøken or boken?
'Boken' (m.) or 'boka' (f.) are the definite singular forms.
This particular noun has an irregular vowel change in its plural forms.
m: en bok - boken - bøker - bøkene
f: ei bok - boka - bøker - bøkene
(a book - the book - books - the books)