A great movie recommendation I'd like to give at this point is 300. King Xerxes is shown in a very villain way in the movie. But Xerxes, Darius, all these Persian Kings are revered amongst the Parsi community.

It's a very occidentalist version of what happened there. Xerxes, his name was Kshayarsh, but the Greeks called him Xerxes.

Modern day Iran is the core of ancient Persia.

It is the successor state of ancient Persia.

I believe it was headquartered in Baghdad.

During Alexander's city of Persia was Parshvapur. Today, the Iranians call it Pasha. They had other big cities under their dominant, like Bagdad, like Damascus as well, by the way. Syria was under Persian control at that time.