Mining difficulty follows price, not the other way around.
If too many miners drop out, the difficulty automatically adjusts so that the block interval is still about the same.
It is elegant in that the more Bitcoin is worth, the more resources are put into protecting it.
This is correct. If the price of BTC increases, then the profitability of mining will increase, causing more people to want to mine and bring miners online. If the price of BTC falls, then mining will be less profitable and less people will want to mine (as it could potentially be unprofitable) so miners will be taken offline.
No, it is not correct, or rather I should say it is incomplete. Difficulty affects price because it impacts miner sales and also the rate at which mined BTC are liquidated to recoup mining costs. It is not now, nor has it ever been, a one-way function.