Learn Russian: Мастерская, в которой выдувают стекло, известна своим изяществом. - The workshop in which they blow glass is famous for its elegance.

This is an indefinite-personal construction. In Russian, when the action itself matters more than who is doing it (similar to the English passive "glass is blown" or the general "they blow glass"), you drop the pronoun они and just use the 3rd person plural verb.

Why use the short adjective известна instead of the long form известная?

Short form adjectives are used as the predicate (the core "is [adjective]" part of the sentence). Furthermore, the short form известен/известна is specifically required when you want to say someone or something is "famous for" something. It connects to the trait it is famous for using the instrumental case (своим изяществом).

Could we say её изяществом instead of своим изяществом?

You should definitely use своим. The reflexive possessive свой refers back to the grammatical subject of the clause (мастерская). If you said её изяществом, it would technically mean the workshop is famous for someone else's (another woman's or feminine noun's) elegance.

Why does the noun мастерская look and decline exactly like an adjective?

Мастерская is a substantivized adjective—a word that started as an adjective (meaning "master's" or "artisan's") but is now used entirely as a noun. Because of this, it takes standard feminine adjective endings in all cases, rather than standard noun endings.