Judah was included in the Roman province of Judea. The entire territory was in Latin, "Iudaea" which literally means "home of the Jews". Iudaea, or Judaea is nowadays spelled Judea.

No. It was the reverse. Jerusalem was a city inside of the Roman Province of Iudaea (Judaea).

The emperor Hadrian changed the name of the Roman province of Judea (also known as Iudaea to avoid confusion with the geographical region of Judaea, Israel) to Syria Palaestina.

There was no Israel in the Roman world. The area that is present day Israel was called Iudaea or Judaea by the Romans. It literally means "the country of the Jews".

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