Every number can be made from the sum of consecutive numbers including prime numbers as prime numbers have only one way of making them. For example, 19 can only be the sum of 9 and 10This partially answers the second question.
The answers to the third question is that all multiples of three excluding 3 can be made with three consecutive numbers. I have even made formulae proving this where:
n is the total
a is the first consecutive number
b is the second consecutive number
c is the third consecutive number
a= 1/3n-1
b=1/3n
c=1/3n+1
n=a+b+c
An example is 12. 12=3+4+5
3=(12x1/3)-1
4=12x1/3
5=(12x1/3)+1
Multiples of 4 may be made with four consecutive numbers although when finding the sum of 4 consecutive numbers, the result is even due to the fact that when two odd numbers are added together, the result will always be even.