The S&P 500 Index is a list of 500 companies maintained by Standard and Poors to roughly represent the US econcomy. That said, being incorporated in the US is not a requirement (like it is with the Fortune 500 list) and as another poster said, there are already 27 companies in the S&P 500 Index that aren't incorporated in the US.
I've owned a couple of companies that re-incorporated out of the US in order to get a lower tax rate. Covidien (COV) was the most recent one, having re-incorporated in Ireland. Nothing happened other than the company was a bit more profitable with the lower rate. The company still traded on the NYSE, the ticker stayed the same.
A big advantage to re-incorporating outside the US (all other things equal) is that in addition to being taxed at a lower rate going forward, all of the money a company has made outside the US in lower tax rate locales can now be used at the corporate level for things like dividends, stock buybacks, and major acquisitions that require a large chunk of money, can now be done without being taxed first when it is repatriated back into the US.
Many companies like Cisco, Apple, Google, Microsoft have tens of billions of dollars "trapped" overseas that they can't bring back to the US unless they're okay with it getting taxed by whatever the difference is between the local tax rate and the US rate. That's one thing I look at when I look at a company's balance sheet. Of all that cash sitting on the balance sheet, how much is actually available at the US corporate level? For example, of Cisco's $52.1 billion in cash, only $4.7 billion is held in US accounts. The rest really isn't available to be paid out to investors unless the company wants to pay tax bringing it back into the US.
As far as Apple trading on the London stock exchange, it could trade there while also trading on the Nasdaq. Stocks can trade on multiple exchanges at the same time. For example, Apple currently trades on both the Nasdaq and the Frankfurt stock exchange:
http://www.reuters.com/finance/stocks/chart?symbol=AAPL.Ohttp://www.reuters.com/finance/stocks/chart?symbol=AAPL.FMike