If I hold shares of a stock on the ex div date, I get a dividend. But if instead I were to short-sell the stock and not buy it back before that date, would I then be responsible for paying the dividend?
Yes, you would. You owe it to the person you borrowed the shares from.
The answer provide by @mbhunter is correct, however there are contexts, shorting in spot market and carrying the position over settlement usually does not entail payment of dividend to the broker, one of the reason being post ex-date the price of the share downward adjusts to the extent of the dividend, so practically if you have shorted at 100 and post ex-date (assuming a dividend of 2 and no movement of the stock price), the price would slide to 98, the party who longed the stock @ 100 now is sitting on a price of 98 and received a dividend of 2 which equates to 100.
The above is also contextual to the law of the country governing the exchange and the security exchange board regulations.
You could hold a long position in some company XXXX and then short your own shares (assuming your broker will let you do that). The dividend that would have gone to you would then go to whoever is holding the shares you short sold. You just don't get a dividend. If you're going to short in a smart way... do it on a stock you otherwise believe in, but use it to minimize the pull-backs on the way up.