To find the median, list the data values in order and identify the middle value. The minutes are already in order: $12,15,15,18,20$. Since there are 5 values, the median is the 3rd value. The 3rd value is $15$, so the median is $15$.

For 5 scores with mean 78, the total of the scores must be $5\times 78=390$. The four known scores add to $72+75+80+83=310$. So the fifth score is $390-310=80$. Therefore, the statement that must be true is that the fifth score is 80.

The diameters in order are already listed as 12, 14, 14, 15, 20. Since there are 5 values, the median is the middle value, which is the 3rd number in the ordered list. The 3rd number is 14, so the median diameter is $14$.

To find the median, first list the numbers in order from least to greatest: 13, 18, 22, 22, 25. Since there are 5 values, the median is the middle value, which is the third number in the ordered list. The third number is 22, so the median number of packages delivered is 22.

The mean is the average value of the data set. When the same number is added to every value in a data set, the mean increases by that number as well. Since each value is increased by 3 and the original mean is 14, the new mean is $14+3$. So the correct answer is $14+3$.