A closed surface encloses charges of +7.0 nC, -1.5 nC and +1.0 nC…

Step 1. The first case. The charge inside is $7.0 - 1.5 + 1.0 = 6.5$ nC. The charge outside adds no net flux. So $\Phi = \dfrac{6.5 \times 10^{-9}\ \text{C}}{8.85 \times 10^{-12}\ \text{C}^2/(\text{N m}^2)} = 734\ \text{N m}^2/\text{C}$.

Step 2. The second case. Now the charge inside is $6.5 + 11.0 = 17.5$ nC. So $\Phi = \dfrac{17.5 \times 10^{-9}}{8.85 \times 10^{-12}} = 1977\ \text{N m}^2/\text{C}$.

Step 3. The field. Yes, the field at a point on the surface changes. The fourth charge makes a field of its own at every point, and moving the charge changes it. Gauss's law tells us only about the net flux, which is a sum over the whole surface.