Modern hebrew gets the name from the Mishnaic hebrew name for the city. It's also spelled with a unvoiced glottal in arabic. Note that Israel was in the hellenistic part of the empire, and caesarea was very much a greek speaking city, where there wasn't palatalization of latin c (which was rendered as a kappa). Note russian Czar.
Turkish also retains the hard C in some forms, the city of Kayseri in Asia Minor is also from Caesarea. However some of that has been eradicated by the more recent influence of French. An example of that is Julius Caesar, which is Jül Sezar.
No... first hebrew is basically just a dialect of Phoenician, and phoenecian isn't an indo european language! It's semitic, like arabic and Akkadian. Also the name Casesarea is of course from Caesar (the city was founded in honor of Augustus by Herod)