Yes, there is no center, that is no contradiction to anything.

Okay, another approach: Imagine you stand on an infinite plane, which has a constant thickness and extends everywhere infinitely. Then no matter where you stand, the plane looks the same regardless of the direction you look. Now imagine this plane is split up in many cubes (like 1m^3 or so), then your gravitational attraction is the same as the sum of the attractions of the cubes. Now make the cubes smaller. And even smaller. In fact, infinitely small. Now you are integrating.