Learn Turkish: Bütün salon bu başarıya şaşakalıyor. - The whole hall is astounded by this success.

It combines the verb root şaş- (from şaşmak, to be surprised) with the compound suffix -akalmak. This suffix shows that someone is "frozen" or stuck in a certain state. So, şaşakalmak means "to be left entirely astounded," as if the audience is frozen in their surprise.

Why is the dative case used on başarıya when the English translation says "astounded BY this success"?

The verb şaşmak (and its compound form şaşakalmak) always takes the dative case (-(y)a/-(y)e) for the thing that causes the surprise. Turkish expresses this as directing your astonishment towards something. Başarı (success) takes the buffer -y- and the suffix -a to become başarıya.

Does salon just mean the physical room here?

Literally, salon means "hall" or "living room." However, just like in English when we say "the whole room gasped," Turkish commonly uses salon to refer to the audience or the people currently inside the hall.

Could I use tüm instead of bütün for "the whole"?

Yes! Both bütün salon and tüm salon mean "the whole hall" or "the entire audience," and they are completely interchangeable in this context.