Is it typically expected that mining difficulty and fiat price are inversely correlated? How much weight does the difficulty have on fiat prices, in your opinions?
None. Higher price drives difficulty*. At one time someone even provide some analysis showing the correlation of price driving difficulty. There is no evidence that difficulty drives price though. The minting rate remains (roughly) the same and today supply is only expanding by 0.3% per day due to newly minted coins. As mining becomes unprofitable the most marginal miners quit and difficulty falls until price rises sufficiently to warrant increased hashing power.
* As the exchange rate rises the profitability of miners also rises. This is because once hardware is bought the primary cost for continued mining is electricity which is priced in fiat. The increased profitability (ROI%) causes more hashing power to be brought online (miners chasing higher profits) and difficulty rises. If price falls the profitability of miners go down and the least efficient miners (marginal miner) go underwater first. Their cost to mine a coin is more than the coin is worth so some of them will idle their rigs and difficulty falls.
