"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.
Они не здесь — literally "they [are] not here". Can't really do anything about the word order, only «не здесь они» is possible but unnatural, any other will be wrong or have different meaning.
Здесь нет их — literally "there is no them". Every part is "separate" so any word order will do, although the nuances will differ as different parts would be stressed in different word orders.
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).
"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