Well, firstly, you need to understand that a Plane does not have a "center point." Planes are limitless, and as such, there are no two edges to measure a center between. The orange square is merely representative and is only to give you a visual clue about where the Plane is. The actual Plane, however, is not limited to that orange square.
There are many ways to make planes.
If you want to define and angle and distance from the Origin, for example, I sometimes sketch a construction line on an existing plane, and construct a new plane-on-a-path on that construction line. Like this:

Then you can control the position/angle of that plane by changing the sketch values (the could be parameters if you like). And if you want stuff sketched on that plane to always relate to a "center point" then when you first start sketching on it, Project in the endpoint of that construction line, and start building the rest of the sketch relative to that projected point as if it were an "origin" of sorts.
If you want the sketch for a new component to always be relative to an existing component, then create an offset-Plane right on the surface of that existing component (zero offset value), or a plane-at-angle right on an edge of that existing component, or whatever (to get the relative orientation you want). Again, when you start the new sketch, the first thing you should do is project in at least one useful point (perhaps the center of your existing hole, in your case) from the existing component geometry, and build the new sketch relative to that point.
Sometimes it is necessary to use an intermediate Plane, and a sketch, on which to base the actual Plane you desire. For example, in my picture above, the sketch, and thus the new plane, was built relative to one of the Origin planes. What if I had wanted the new Plane angled somewhere BETWEEN the X and Y origin planes? I might have had to do something like this:
