The Phoenicians were indeed a distinct people who spoke a language in the Semitic language family and originally inhabited the area around the cities of Tyre, Sidon, and Byblos, in modern-day Lebanon. There was no Phoenician state, but rather a loose confederation of independent city-states and, later, colonies. The Phoenicians were skilled sailors and traders who founded many colonies along the coasts of North Africa and Iberia as well as several Mediterranean islands, like Sardinia and Sicily. Carthage was one of the most important Phoenician colonies, and it was so powerful that Rome finally felt compelled to raze it to the ground after the third Punic (a word that derives from "Phoenician") War. Phoenician and Hebrew are both part of the Semitic language family but Phoenicians developed the alphabet first. Neither could accurately be called a "subset" of the other. Unfortunately, I parted with my books on ancient history some time ago, so the only reference I have on hand is Margaret Oliphant's The Atlas of the Ancient World!

From what I've read Phoenicians were not Israelites, but were related to them as both are considered Semitic people. As far as your second question is concerned, it seems Phoenicia was an appellation used by outsiders to refer to the general region/civilization of Phoenicia but the Phoenicians themselves did not hold themselves to a over-arching Phoenician identity and instead identified mostly with the city state they were living in. Each Phoenician city-state (not too different from the political situation in Greece) was a power in its own right, and had its own unique culture, although still similar enough to each other that an outsider would consider their inhabitants to be part of the same people. So that's the short answer.

source: The Phoenicians: The History and Culture of One of the Ancient World's Most Influential Civilizations