Help me to restore this beautiful music box

Hello! I added this masterpiece to my collection as I bought it very cheap. It has a lot of damages but I will give it a try. It is a carousel lipstick/cigarette music box and I think that it is dated at 1960s. As it comes to the metal parts I will take them to a jewellerist to polish them and gold plate them where it needs. I want your help to the marble? Glass? Parts. They have a light blue coat. At first I thought it was glass with a marble coat effect but finally I think it is white marble with a blue coat. There are some missing marble parts at the door and I will make new at a marble technician but I need your help at the coat. I have never seen a finish like this on a marble. Does anyone know what kind of coat it is?

Or any other information that you have about it, would be very helpful.
Thank you!

 

Here are some similar and a same one as mine to see how it should be.

 

Looks more like alabaster with a tinted blue lacquer coating.

 

Here are some of yours posted full size for easier viewing for all!! I think the next to last image posted shows best what @verybrad was talking about! Looks like you have your work cut out for you!!! Good luck!!









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Just my subjective opinion, but I see glass with trompe l'oeil painting. Evidently these were mass produced. Molded glass would be more efficient than grinding, polishing & tinting stone of any kind.