"Lover" can't be used here?
Is it Boyfriend because it ended in an -en instead of an -e?, because it accepted girlfriend. Are both right?
No. "kjæreste" is not a gender specific word. It ends in "en" because that's the ending for definite and singular masculine nouns. It can be both a female or a male.
In that case I don't understand why "his boyfriend" seems to be prefered over "his girlfriend".
It's not - both are accepted.
Is it not possible for the translation to say "His boyfriend/girlfriend...". I'm all for equality, but as the person said above, it is more confusing by just assuming that is the translation.
i think that instead of '' ham'' it can as well be used ''han'' here. Am i right?
Yes, both can be used as an object pronoun.
However, we're only able to accept the one the TTS thinks she's actually saying for the listening exercises, which is always "ham" as far as the object pronoun is concerned.