Learn Russian: Здесь они строят большой дом.

The infinitive of this verb is строить (to build). Because it ends in -ить, it belongs to the second conjugation. For они (they) in the second conjugation, verbs drop the -ить and add -ят (or -ат after certain letters). If it were a first conjugation verb, it would usually take -ют or -ут.

Is большой дом in the nominative or accusative case here?

It is in the accusative case because it is the direct object (the thing being built). However, for masculine inanimate objects (non-living things like a house), the accusative form looks exactly the same as the nominative (dictionary) form. That is why дом and its adjective большой do not change their endings here.

Does строят mean 'they build' or 'they are building'?

It means both! Russian does not have a separate continuous tense like English does. The present tense verb строят covers both general facts ('they build') and actions happening right now ('they are building'). The context tells you which one makes sense.

Could I put Здесь at the end of the sentence?

Yes, you could say Они строят большой дом здесь. Russian word order is very flexible. However, putting Здесь at the beginning sets the scene immediately, which is a very natural and common way to structure sentences in Russian.