Does each stock exchange only trade in one currency? And if not, how are currency exchange rates accounted for?
An exchange can handle more than one currency, but this would be uncommon for equities.  An example is the ICE futures exchange in Europe. 

Multiple currencies doesn't cause any particular problems - transactions are carried out in currency and it is up to you to convert to whatever currency you prefer.  Usually done by fx sweeps by your custodian bank.

For margin purposes combined exposure (e.g risk netting) can be calculated at prevailing fx rates.

Each security only trades in one currency, and each security trades basically like a mini exchange so there isn't really any problem.  Think of it like two exchanges run by the same company.