Learn Russian: Невзирая на хрупкость, мастер не разбил стекло. - Despite the fragility, the master did not break the glass.

To use that construction, you would say При всей хрупкости стекла, мастер его не разбил (Despite all the fragility of the glass, the master didn't break it). Remember that при всей takes the Prepositional case, which is why хрупкость changes to хрупкости.

What case does невзирая на take, and why hasn't the word хрупкость changed its ending?

The preposition невзирая на requires the Accusative case. Хрупкость is a feminine noun ending in a soft sign (a 3rd declension noun). For inanimate nouns in this category, the Accusative form is exactly the same as the Nominative.

Is невзирая a verb? It looks like a gerund (verbal adverb).

Historically, yes! It comes from the verb взирать (to look at), so literally it means "not looking at". However, невзирая на has frozen into a compound preposition meaning "despite", acting just like its slightly more common synonym несмотря на.

Does мастер mean "master" in the sense of a boss or a martial arts master?

Usually not in this context. In everyday Russian, a мастер is a skilled craftsman, technician, or repair worker. Here, it most likely refers to the glazier or the skilled workman who was hired to install or handle the glass.