Lisa Said:

*^*^*^* DIGIT DISCOVERY *^*^*^*?

We Answered:

The digits that are non-prime (column 2) are 0, 1, 4, 6, 8
The digits that are non-composite (row 2) are 0, 1, 2, 3, 5, 7

That means the digit in the center has to be 0 or 1. But a clue says all numbers in row 2 are odd. So that means it must be 1 in the middle
- - -
- 1 -
- - -

The number zero is the only digit that is not a counting number (1, 2, 3, etc.). Thus by the clue it can't appear in column 2 or 3, or row 3. And we already know it can't be in row 2 because it only has odd numbers. That leaves the top left corner.

0 - -
- 1 -
- - -

Row 2 can only contain odd numbers, so the remaining possibilities are 3, 5 and 7. But we are also told it adds up to 13. That means you must have 5 and 7 in that row. You are also told that the sum of the first column is less than 8. If you put a 7 there you'd need a zero again to make the sum less than 8. That's not possible because 0 is already used, so it must be a 5.
0 - -
5 1 7
- - -

And for column 1 to be less than 7 you must have a 2:
0 - -
5 1 7
2 - -

The remaining choices for the last row must be consecutive even integers so that means 4 and 6, but we don't know the order. That leaves digits 3 and 8 to go in row 1. Because column 2 can't contain a prime, you have to have an 8, which puts the 3 in column 3.
0 8 3
5 1 7
2 - -

The first row adds up to 11 which is the same sum as the corners. So the last corner must be 6. And that then determines where 4 goes.

Answer:
0 8 3
5 1 7
2 4 6