Short answers, pulled from the story.
Julius Caesar introduced the Julian calendar in 45 BC after realigning the start of the year to the 1st of January. The reform required making 46 BC a massive 445 days long to compensate for missed intercalations during his pontificate.
Sosigenes of Alexandria served as the principal designer of the Julian calendar reform alongside Greek philosophers and mathematicians. The new calendar began operation after the realignment completed in 45 BC following the insertion of two extraordinary intercalary months between November and December.
The Julian calendar has an average length of 365.25 days which exceeds the actual tropical solar year value of approximately 365.2419 days. Consequently the calendar gains about three days every four centuries compared to observed equinox times.
Pontifices initially added a leap day every three years instead of every four after Julius Caesar's death creating twelve years without leap years between 9 BC and AD 4. Emperor Augustus intervened to fix the triennial leap year error and restore the four-year cycle by AD 8.
Spain and Portugal adopted the Gregorian calendar in 1556 while Prussia and Denmark-Norway did so in 1559. Greece was the last country to adopt the Gregorian calendar as its civil calendar in 1923.