Achilles (Greek Ἀχιλλεύς Akhilleus) is a character from the epic poem called the Iliad. We do not know when the Iliad was composed, but we know that it was widely known in Greece by the seventh century BC. In the Iliad, he is not invincible; he fears death like every other warrior, and is even wounded in the arm during his rampage after the death of Patroklos. There is no reference to him being dipped in the river Styx as a child to make him impervious to weapons. That story first appeared in writing more than 700 years later, in an epic poem called the Achilleid, by the Roman Statius. So it's pretty safe to say that the name Achilles existed long before that name became a reference to any particular feature of human anatomy. The tendon was named after the Homeric hero, not the other way around.

As for Paris (Greek Πάρις Paris, also Ἀλέξανδρος Alexandros), there is no relation between his name and the city in France. The hero's name is older; while the Iliad goes back at least to the early 7th century BC, there is no archaeological trace of the city that would later be called Paris until the 3rd century BC. But in any case the city was not named after the hero. In Roman times, the town was known as Lutetia. It was later renamed Paris after the Celtic tribe of Parisii that originally settled the place.