You have it backwards.
No, you do. Do you really think that when if people see that the difficulty increase, they think "wow, now I've got to buy hundreds of thousands of dollars worth of bitcoins, so the price keeps up with the difficulty". The miners consider the price, and if they think it's profitable enough to mine they will buy more mining hardware, which will push up the difficulty.
Just look at the difficulty after the price sky rocketed to $1. The difficulty jumped from about 30 000 to about 75 000 in three jumps, and then pretty much leveled out while the price fell back.
Absolutely. I believe that the mining difficulty increase signals to people that BTC will become more valuable because BTC will be mined at a lesser rate for the same hardware in USD. This requires BTC to increase in value over the USD because BTC will cost more in USD to produce. Smart traders recognize this, and realize the FV of BTC makes it well-worth buying up BTC at its current prices because difficulty will continue increasing, increasing the USD cost to produce BTC, and thus leading to an increase in the price of BTC.
Consider this: You produce toys (or, BTC). Let's say commodities (may help to think of them as video cards) to produce toys is limited (for example, only 21m toys can ever be produced with the Earth's reserves of resources). As commodities become more and more scarce, and thus more difficult and expensive to harvest, they will increase in price. The toy-makers will have to respond by increasing their own prices to keep a decent profit margin while cost of production increases. Traders, with knowledge that toys (and the commodities to produce them) will become increasingly difficult and expensive to produce, buy up the toys and commodities creating a speculative bubble where toys become outrageously expensive to anyone not holding or producing the toys. However, those trading and producing toys know commodities are ever-still becoming more scarce, and continue buying up toys & commodities to produce toys, creating a self-contained market where kids looking to actually use the toys either fork over outrageous sums of money, or just decide against using toys.