not sure about glutSphere and texture coordinates. if you roll your own: there are two approaches to a sphere - implicit surface and parametric. generally parametric surfaces (NURBS, uv space, etc. ) are used because they are easier to span in finite, user determinable steps – a key feature to have since you’ll be required tessalate yourCustomSphere(). tesselation is a simple process but many many people have worked on it – it’s a large topic.
i believe opengl provides routines for tessalating parametric surfaces, but they are presented only briefly in the red book. might i suggest using a third party tool (such as Maya or 3DSMax, et al) to create geometry, then importing it. the .obj file format is popular. a bare bones loader is included in the jogl demo source (utils), but you’d have to add the texture support. it currently only reads vertices and normals.
anyone here built there own tesselation routines for parametric surfaces? love to hear about it.