A3 = A1 + 2d
A7 = A1 + 6d

Given that A3 + A7 = 5:
A1 + 2d + A1 + 6d = 5
2A1 + 8d = 5
A1 + 4d = 2.5

Given that a4 = 1:
A1 + 3d = 1

Subtracting the two equations above:
A1 + 4d = 2.5
A1 + 3d = 1

d = 1.5
A1 = -2

The arithmetic progression would be: -2, -0.5, 1, 2.5, 4, 5.5, 7, 8.5, 10, 11.5

Sum of the first 10 terms = 46

For the inequality 0 < A1 + 9d:
-2 + 9*1.5 = 11.5

There are 12 integers between 0 and 11.5, so the sum of all the integers would be 78.