What's the reason that we have a different number of days each month? It always was interesting for me to understand the answer for the question:


  What's the reason that we have a different number of days each month?


If the month is fixed on the time that the moon circles the earth, then the time of the circling isn't equal each month? And if the month fixed by the time that the earth circles the sun and then divided by 12 (365:12=30.4), why one month has more days and the other month has less days?

Before desktop computers, how did astronomers handle calculations of dates and times accurately, for example two plates taken decades apart, how would an astronomer calculate the time difference precisely?
You make a great point. The reason behind the discrepancy between the dates is due to a complicated history behind it.

The calendar is based on the calendar created by ancient Romans, which is based on one Moon cycle. One lunar cycle is 


  29.53 days.
  
  
    www.universetoday.com/20620/lunar-year/
  


which does not evenly divide into the 365.25 days of the year.

Note: The Romans borrowed the calendar from the ancient Greeks to develop the idea of a 10-month calendar that left approximately 60 days unaccounted for.

Consequently, the earliest ancient Roman calendars had months that were either 29 or 30 days to account for this discrepancy. 

The Romans started using the 10-month calendar in 738 B.C. Their months were initially called:


Mensis Martius (March)
Mensis Aprilis (April)
Mensis Maius (May)
Mensis Iunius (June)
Mensis Quintilis (July, quin- meaning "five")
Mensis Sextilis (August, sex- meaning "six")
Mensis September ("seven")
Mensis October ("eight")
Mensis November ("nine")
Mensis December ("ten")


To account for the remaining $\sim60$ days, Mensis Ianuarius (January) was added to the beginning of the year and Mensis Februarius (February) to the end of the year during Numa's reign around 700 B.C. with a leap year every 4 years. 


  Leap years were decreed by proclamation the pontifex maximus, in such years an extra month was added to the calendar.
  
  
    Quoted from PM 2Ring
  


The calendar stayed in that order until 452 B.C. when a small council of Romans, called the Decemvirs, moved February to follow January. 

However, 


  in the initial 36 years of its adoption leap days were added every 3 years due to a misunderstanding (Romans used inclusive counting),
  
  
    Quoted from PM 2Ring
  


so that caused a big problem. In fact, by the start of the reign of Julius Caesar, the previous calendar was off by a whole week!

Julius Caesar noticed this and thus modified the Roman calendar in 46 B.C. to make each month have either 30 or 31 days, with the exception of Februarius[February], which had 29 days, with the three year leap year fixed into a four year leap year. Quintilis[July] was later renamed Julius[July] in his honor. Likewise, Sextilis[August] later became Augustus[August] to honor Augustus (Gaius Octavius, the first Roman Emperor). Augustus[August] was also given an extra day (taken away from Februarius[February]), so that Augustus and Julius would have an equal number of days, representing their equal power. This is the Julian Calendar.


  This calendar was extremely accurate, but was still not accurate
  enough as it drifts by approximately 3 days every 400 years. So after
  a few hundred years, the drift becomes noticeable
  (i.e. the seasons don't properly match the calendar), which is
  important for farming, and therefore human survival. 
  
  
    quoted from  CJ Dennis
  


Then a pope named Pope Gregory XIII modified the Julian calendar again in October 1582. This is the currently used Gregorian calendar. He made it so that 


  Every year that is exactly divisible by four is a leap year, except for years that are exactly divisible by 100, but these centurial years are leap years if they are exactly divisible by 400.
  
  
    https://en.wikipedia.org/wiki/Gregorian_calendar
  


This calendar is the most accurate calendar currently used, and won't be a single day off until the year of $\pm 3200$ AD. It isn't perfect though. Compared to the tropical year, it is 27 seconds too long. This is...


  due to astronomical effects (the
  rotation of the Earth speeding up and slowing down, etc.) we can't be
  more precise than that. 
  
  
    quoted from CJ Dennis
  




As for your other question, unless the plates taken had accurate dates of when recorded, it would be rather impossible to calculate precisely.