Learn Italian: Hanno trovato un tempio nella Grecia antica. - They found a temple in ancient Greece.

Normally, singular country names take just the preposition in without a definite article (like vado in Grecia). However, when a country is modified by an adjective or a descriptive phrase, the definite article becomes mandatory. Because Grecia is modified by antica, in must combine with la to become nella.

Could we put the adjective before the noun and say nella antica Grecia?

Yes, but it would contract to nell'antica Grecia. Placing the adjective before the noun gives it a more evocative or literary feel, which is extremely common when discussing classical history. The phrasing nella Grecia antica used here is simply more neutral and descriptive.

Is there anything tricky about the plural form of tempio?

Yes, the plural is templi, gaining an unexpected l. You have to be careful not to write tempi, because that is the plural of tempo (meaning time or weather).

Since this is about ancient history, shouldn't we use the passato remoto instead of hanno trovato?

The temple itself is ancient, but the action of finding it (likely by modern archaeologists) is recent. You only use the passato remoto for events that happened in the distant past and have no connection to the present. If you were describing the ancient Greeks building the temple, then the passato remoto would be required.