During the life of Cato the Elder, was Latin spoken with hard or soft "c"s?

Answer 1

Hard. The shift away from the hard C is post-classical, and part of how we know this is the way that Greek authors transliterated Roman names. Even Appian, who lived in the 2nd century AD, transliterated "Cato" into Greek with a kappa, which had (and even does today in modern Greek) the hard "c" sound: Κάτων (2.14.98).