As far as testing goes, I would suggest using the second slice down in picture 2. Separate and flat polish a sample of just the core material, and test its refractive index and specific gravity. These two tests should greatly narrow down the field as to what the core material may be comprised of.
The seed may not be chalcedony at all. All that the seed needs to do, is provide a surface for the chalcedony to anchor onto. This can be almost anything present within the growing environment. Even dinosaur bone has been used as nature's seed material.
It would be helpful to know the origin of the specimen. That may help inform to the seed composition.
What I find interesting, is how "egg-like" that nodule appears to be. While a "petrified egg" is a flight of fancy, Agate forms of Chalcedony are known to petrify wood, and Silicified dinosaur bone has been discovered. An analysis of the structure of the core may provide some interesting insights.
There are so many different forms of chalcedony, that narrowing it down to a single variant would be difficult. Agates have translucency while jasper is opaque. Both are forms of chalcedony. And then there are hybrids referred to by some has jasper-agates.
The grayish outside is suggestive of chert, and the core could be Aventurine. (Opaque, compact Quartz / Chalcedony containing small Mica, Hematite, or Goethite scales which cause a glistening effect. Aventurine is most often green but may also be other colors such as gray, orange, and brown. -source- http://www.minerals.net/gemstone/chalcedony_gemstone.aspx). I am not able to see if there is any shimmer from the images.
All chalcedony will have similar testing results. The only way to truly identify the core, is if it comes back as something other than chalcedony.