Do transaction fees remain stable in bitcoins, or adapt to purchasing power? 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).