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.
This is an example of an exchange trading a bond in EUR even though the normal trading currency is DKK: http://www.nasdaqomxnordic.com/bonds/denmark/microsite?Instrument=XCSE2NYK13HJ20LRT
Nothing stops the stock exchange from denominating equities in different currencies and settling trades in different currencies.
The Singapore Exchange lists equities denominated in different currencies, including Singapore dollars (SGD), US dollars (USD), Hong Kong dollars (HKD), Renminbi (CNY), Japanese Yen (JPY), Euros (EUR), Australian dollars (AUD), Pound sterling (GBP).
If you want to buy or sell a stock that happens to be denominated in USD, then you must settle the trade using USD. Most brokers in Singapore are able to give their clients multi-currency accounts. For example, if you want to buy a stock denominated in USD, then you can either use funds from your USD account (if you have one), or ask the broker to convert funds from your SGD account into USD. Brokers set the exchange rate for currency conversions, and their trade commissions may differ for stocks denominated in different currencies.
In 2012, dual-currency trading was introduced on the Singapore Exchange. A company can choose to list its securities in two different currency denominations (e.g. SGD and USD, one stock ticker for each currency). For these dual-currency listings, an investor can trade the security in any of the two currencies regardless of the currency in which it was first bought or sold. So if you own shares in a company with a dual-currency listing, you can sell your shares in any of the two currencies.