Is there any merit to the theory that Dorne deals with slavery and that is the source of the Martell’s opulence? Is Areo Hotah a slave?

No, I don’t think so. Slavery is a taboo in Westeros  – “the old gods and the new alike hold slavery to be an abomination” – and given how the Rhoynar suffered at the hands of slavers, both in the wars against Valyria and during Nymeria’s wanderings, I doubt they felt any different.

Areo Hotah was a temple slave in Norvos during his youth, but almost certainly was manumitted when Lady Mellario was married to Doran Martell and moved to Dorne. Indeed, as evidence of his freed status, we can point to the fact that Areo Hotah chose to stay and serve Doran Martell when Mellario separated from her husband and moved back to Norvos.