Is there a science to calendars?

You are correct in that there are 14 different annual calendars for the most commonly used one in the western world and commerce. And there is a simple pattern of their use from 1901 to 2199 because the except rule applied to 2000 and it was a leap year. As long as you use the right non-leap year calendar for 1900 and 2100 you can keep on going.

Early versions of Visicalc and Lotus 123 got some date calculations wrong because they gave 1900 a leap year it did not have. Excel has a quirk to match (year 0)

Certainly. It was last studied in detail by a committee commissioned by Pope Gregory XIII in 1582. This created what we call the Gregorian Calendar, which almost all the wold uses today. There are many more than 14 ways the calendar can fall, because of the complex rules of leap years.

I assume you mean the standard civil Gregorian calendar. The full cycle repeats every 400 years, which is exactly 20,871 weeks. There aren't any special names for the years, except "leap year."

For common years, the number of years starting on each weekday is 43 (Monday), 44, 43, 44, 43, 43, 43 (Sunday), for a total of 303 common years per 400 year cycle.

For leap years, the numbers are 12 (Monday), 14, 14, 13, 15, 13, and 15 (Sunday), for the 97 leap years per 400 year cycle.