The Bitcoin price now is about $660 per BTC, and if I send some, the transaction fee will be about 0.0001 BTC or roughly $0.067. When the Bitcoin price moves to $1200 per BTC, would the transaction fee still be 0.0001 BTC, i.e. then $1.79?
If the price goes to $12000 per BTC, and I just want to sent $5 to $10, would I also need to pay the such a high transaction fee?
Your understanding is correct. If the required fee rate on the network remains the same (denominated in BTC) and the value of BTC rises against USD, then the transaction fees rise in USD terms.
In reality, there are many factors which affect what the actual required transaction fee will be, such as time/day of transaction (there's generally less competition for block space over weekends), size of transaction and if the transaction uses segwit (since segwit witness data is discounted).
The transaction fees in Bitcoin are not directly tied to either the amount of BTC you are sending, nor the current value of BTC. Instead, they are market-based and charge by the byte. If paper checks were like Bitcoin, the processing fee would be determined by the size of the paper the check is printed on...not the amount.
The market-based aspect comes from the limited space in a block. For your transaction to make it to a block, it has to be more worthwhile to the miner than other transactions. Remember that miners are essentially the gate keepers of the scarce resource that is each block. Also remember that they pay their electricity bills in fiat, but make their money in BTC. Therefore, if the price were to drop, and the number of transactions waiting to be confirmed increase, we would likely see a rise in transaction fees, even when converted into fiat. This is because miners would become more selective over which transactions make it into a block, choosing only those with higher fees.
