Arbitrage should theoretically close this gap in the longer run. A Canadian study found a lack of arbitrage opportunity in closing prices over a month, which would indicate that shorter term arbitrage opportunities are being closed, as one would expect. http://www.coindesk.com/bank-canada-research-cryptocurrency-arbitrage-doesnt-exist/

Are you sure these price differences you're seeing can't be explained by exchange fees? I wonder if the differences in price, up to the point that arbitrage becomes unprofitable, would indicate a country's relative interest in BTC since one currency must have a higher relative value?

A fiat-index as you've suggested should be unnecessary, if not now then when the market matures.