Why are foreign Cos listed on the US exchanges?
Liquidity. The american stock exchanges are the largest in the world. A bigger, more liquid market is considered to be a positive thing for any stock.
Since most Americans wouldn't like the complications involved with holding foreign companies' stocks directly (currency fluctuation risk, weird foreign stock exchange rules, foreign tax withholdings, etc.), so Wall Street created what are called ADRs (American Depository Receipts), which avoid all those problems. They trade in American dollars, on an American exchange, and follow all NYSE and SEC reporting and other rules.
Voila, you can trade Daimler-Chrysler just as easily as you can Ford Motors now.
Can they be listed on every countries exchanges?
Theoretically, it they don't mind paying the listing and other costs associated with being traded on a particular exchange. Trading on the NYSE is a no-brainer, but would Daimler Chrysler be a hot selling stock on say, the Tapei Exchange? Only the company can answer that question.
Is DCX the same stock you'd buy in Germany for the company?
Essentially, yes, only you'd be paying in Deustchemarks. Since most Americans have dollars instead of marks, they generally buy the ADRs rather than the underlying stock on the home exchange.