To get the centroid you can use Vector->Geometry Tools->Polygon Centroids. Then to get the coordinates you can either add a couple of fields to the attribute table and use the field calculator set to $x and $y respectively for each field or use the Coordinate Capture plugin.
To get the maximum distance you will need to do a little more work. A very simple approach would be to convert your polygon to nodes (Vector->Geometry Tools->Extract Nodes) and then use the distance matrix tool (Vector->AnalysisTools->Distance Matrix). This will give you the distance to every vertex of the polygon and you merely need to sift the result to get the highest value. To improve the accuracy of this approach you could densify your vertices first. If you ensure that your centroid and nodes point datasets preserve an ID from the polygon, this operation should be straightforward.
You could also use the PostGis ST_Distance_spheroid function instead of the QGIS Distance Matrix function as this will give you the linear distance on a spheroid and possibly save you reprojecting your data. Unfortunately the ST_Distance function only gives you the minimum distance so either way, you will need to iterate all the points for a given polygon.