"It seems to me, they are not here."
Translation:Мне кажется, они не здесь.
Yes, it's equally correct with «что». Please report this sentence when you get it again.
"To seem" казаться has an interesting conjugation - it seems to act pretty much like other verbs, except the -ся at the end doesn't change except for 2nd person plural, where it becomes -сь
1st Person Singular - я кажу́сь - 2nd Person Singular - ты ка́жешься - 3rd Person Singular - он ка́жется - 1st Person Plural - мы ка́жемся - 2nd Person Plural - вы ка́жетесь - 3rd Person Plural - они ка́жутся
This is a great example of the way negation works in Russian. Two correct versions of this are: Их здесь нет: Not they [genitive case] are here. Они не здесь: They [nominative case] are not here.
While the effects of negation are a lot more complicated than this, it seems like не has a far less broad scope than нет.
«Нету» is a colloquial form (in literary language, we only use «нет»). Also, your version doesn't translate 'here' from the original sentence.
«Кажется, их здесь нет» should work I think (although I'm not sure if Duolingo accepts it).