Hint: There are two types of $4$ digit odd integers with all digits different: (i) First digit is even ($2,4,6,8$) or (ii) first digit is odd. Count the Type (i) numbers, the Type (ii) numbers, and add.
Type (i): There are $4$ choices for the first digit. For each of these choices there are $5$ choices for the last digit. For each of the choices we have made so far, there are $8$ choices for the second digit, and for each such choice $7$ choices for the third, a total of $(4)(5)(8)(7)$.
Now it's your turn: make a similar calculation for Type (ii).
The analysis for the even numbers will be roughly similar. The two cases are (i) last digit is $0$ and (ii) last digit is not $0$.
Remark: The tricky thing is that the first digit cannot be $0$. A slightly nicer approach, I think, is to first allow $0$ as a first digit, and then take away the "numbers" with first digit $0$, which we should not have counted.
So for odd numbers, if we allow $0$ as a first digit, we can choose the last digit in $5$ ways, and then for the rest we have $(9)(8)(7)$ choices, for a total of $(5)(9)(8)(7)$.
Now if we have $0$ as the first digit, the rest can be filled in $(5)(8)(7)$ ways.
So our answer is $(5)(9)(8)(7)-(5)(8)(7)=2240$.
For the evens, the same analysis gives $(5)(9)(8)(7)-(4)(8)(7)$.
Given the simple facts that even+even=even, even+odd=odd, and odd+odd=even, we can see that if there are an even number of odd digits, the sum of the digits is even and if there is a odd number of odd digits, the sum of the digits is odd.
So, rather than summing the digits, you can simply count the number of odd digits, which has the same parity as the number itself.
Best Answer
As straightforward mathematics there is no answer.
As anyone who has ever placed hymn numbers in a hymn board will know, it is possible to turn $9$ upside down to get $6$, and if this is allowed by the wording you can get a sum of $30$.
Likewise if it is odd numbers which are chosen, but the digits rather than the numbers which are added, the set $3,5,7,9,15$ gives $3+5+7+9+1+5=30$ - again this depends on precisely how the question is worded.