The smallest is 14.
Not possible in consecutive integers, nearest is consecutive even integers: 148 & 152
It is not possible in our Universe that two consecutive numbers can total an even number, since any two consecutive numbers must be an odd and an even (or vice versa). There are two consecutive odd numbers which total 132 ie 65 and 67
The smallest of seven consecutive even integers whose sum is 700 is 94.
There are two consecutive even numbers. The numbers are 92 and 94.
There are two consecutive even numbers. The numbers are 62 and 64.
There are two consecutive even numbers. The numbers are 26 and 28.
Consecutive even numbers are any even numbers which are separated by one (uneven) number... so that 6 and 8 are consecutive even numbers... as are 22, 24 and 26. Similarly, consecutive odd numbers are numbers which are separated by one (even) number... so that 11 and 13 are consecutive odd numbers... as are 31, 33 and 35.
Defining "consecutive" as "following continuously in unbroken or logical sequence," it is possible to have many different types of consecutive things: consecutive days, months, odd numbers, even numbers, etc. The list you have is consecutive, they are consecutive multiples of ten.