Price influences difficulty for certain. There are a bunch of other factors, but for the most part this holds true (currently):

-People want to mine when the price rises.
-People can't afford to mine when the price drops.

The reverse hypothesis (difficulty influences price) is probably way less of a significant correlation. Other than the fact that an acceleration in hashrate growth - like we are seeing now - causes more bitcoins to be released than the anticipated 1block/10min. Those "extra" bitcoins are being put in the hands of people who likely just spent a bunch of money on new mining equipment, and want to sell the coins to offset their initial investment. Thereby increasing sell volume at strike.

You can see how many "extra" coins are being created per day here: http://blockchain.info/stats


As for alternative hashing algos, if it ain't broken...


tl;dr Price affects hashrate, hashrate doesn't affect price.