If Caesar did crunch the numbers himself, where did he get his mathematical education? If not Caesar, who did design the calender? Were there are attempts to reform the calender before Caesar?
No, it was an astronomer Sosigenes who designed the reform.^(1) But other sources on the reform^(2,3,4,5,6,7,8,9,10) attribute the reform to Caesar personally, so it's pretty clear that he had an interest in astronomical matters himself.
The use of a solar calendar was apparently inspired by the model of the Ptolemaic Alexandrian calendar,^(2,3) which was solar (the republican Roman calendar was basically lunar, though thoroughly warped by Caesar's time). The state of Ptolemaic astronomy was certainly sophisticated enough to have been easily capable of determining the correct length of the solar year for some centuries. For this reason it seems likely that Sosigenes was an alumnus of Alexandria.
(Edit: I had this wrong. The Alexandrian solar calendar was instituted after the Julian calendar -- probably about 30 BCE. Oopsie.)
Unfortunately, we know nothing about Sosigenes beyond the brief mentions in Pliny the Elder^(1) (NB: he should not be confused with a 2nd century peripatetic philosopher of the same name).
The most specific and, it seems, accurate account of the reform comes from Censorinus (Latin text):^(7)
(The calendar) became so inaccurate that in order to correct it, C. Caesar, during his 3rd consulship along with M. Aemilius Lepidus (i.e. in 46 BCE), inserted two intercalary months of 67 days between November and December, after he had already given February an intercalation of 23 days. This made the year 445 days long. At the same time, to prevent the same error happening again in future, he disestablished the intercalary month and shaped the civil year in accordance with the sun. So to the year of 355 days he added 10, which were distributed among the seven 29-day months to give January, Sextilis, and December 2 days each, and the rest 1 day each. He placed these days at the ends of the months so that religious festivals should not be displaced from their position in the month. As a result, while we now have seven months of 31 days, the four months that had been originally appointed to have that number are distinguished from the others by having the Nones on the 7th; the other three have the Nones on the 5th. And to account for the quarter day that was required to complete a true year, he established that after the course of each four years, 1 day was intercalated after the Terminalia festival (23 February), which is now called "bis sextum" (i.e. a second 6th day before the Kalends).
And here's what Pliny has to say (tr. Bostock; Latin text):^(1)
To these has been added a fourth school, which was established by the Dictator Caesar among ourselves, and to which was entrusted the duty of regulating the year in conformity with the sun's revolution, under the auspices of Sosigenes, an astronomer of considerable learning and skill. His theory, too, upon the discovery of certain errors, has since been corrected, no intercalations having been made for twelve successive years, upon its being found that the year which before had anticipated the constellations, was now beginning to fall behind them. Even Sosigenes himself, too, though more correct than his predecessors, has not hesitated to show, by his continual corrections in the three several treatises which he composed, that he still entertained great doubts on the subject. ...
^(1)Pliny Natural History 18.211-212 (ยง57)
^(2)Appian Civil War 2.154
^(3)Dion Cassius 43.26
^(4)Plutarch Caesar 59
^(5)Ovid Fasti 3.155
^(6)Suetonius Julius 40
^(7)Censorinus 20.8-10
^(8)Macrobius Saturnalia 1.14
^(9)Ammianus Marcellinus 26.1.8-14
^(10)Solinus 1.45