The idea of currency was to establish a medium of exchange that can be exchanged and accepted for anything. Gold was chosen to become the first currency in a shape of a bean, later flattened into the circular disk we use as coins today. There was enough gold available, and desired by everyone, the government of Lydia mint these electrum 'coins'.

Barter is not technically replaced, it is just called buying. A person would still 'barter' their coins for a pig or shoes or whatever they desired.

Lydian King Alyattes decided to mint electrum (a gold and silver alloy) coins because gold and silver had already started to emerge as the most exchangeable good. King Alyattes hoped that by minting coins bartering would become easier because the coin would have a set weight which would equal a set value (this value of course fluctuated with the price of gold).

Which brings us to your last question. Nations were able to trade currencies because the currency had intrinsic value, it's composition, its weight of its metal material was the value of a coin. Those first Lydian coins were literally nuggets of metal stamped with a face of a lion and a stamp stating it weight. Not like todays money where a dollar=dollar because the government tells you it is.

The best two books to read about the rise of money would be

Electrum and the Invention of Coinage by Joseph Linzalone

The Invention of Coinage and the Monetization of Ancient Greece by David Schaps

But also if your interested check out Debt by David Graeber because although it is not an overly scholarly book it was a fun and interesting read.