It's handpainted and gilded.
Dating would be a guess. I'd have said c. 1860s but that is a guess. You'd need to find a similar shape with similar base to verify that.
The interior is water damaged. I am pretty sure it hasn't been coated with anything,
The alabasterglas is just thinner at the rim because it is. Not for any reason.
It's white alabasterglas hand enamelled and gilded. It just happens to have a greyish/blueish cast to it.
If I get onto JStor at the end of the month when I get my allocation, and manage to read the article, and it has anything interesting linked to your piece, I'll come back

There are no other pieces in the museum that would help.
You could try searching Dr Fischer Auctions, also Glas Kilian and Antiques Neurwirth who all have a very good collection of Biedermeier and Mid 19th century glass.
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