Questions about the gold standard and exchange rates
I would interpret the “Gold Standard” as being all (major) countries each pegging their currency to gold, or at least to a currency pegged to gold (e.g., the U.S. dollar in the Bretton Woods system). A single country pegging its currency to gold would make gold investors happy, but it would do very little otherwise, since the currency would just float like gold prices already do.

This means that the exchange rates cannot appreciably move (there is usually a small trading band around parities). 

This is why that DeLong says that the decline “... would not be allowed.” That is, if tge currency was weakening within its small trading band, the central bank needs to defend the parity: typically raising interest rates. If that does not help, the government is forced to launch fiscal austerity, which shrinks the economy. This reduces imports, which will hopefully take the pressure off the exchange rate.