Now back to your question, yes it's true as the price of bitcoin increases, gas fees increases too. There's no avoiding that.
That is not correct, it is transaction fee and not gas fee, BTC network uses transaction fee, while the Ethereum network uses gas fee. TX fee isn't affected by the price of BTC, the exact transaction fee you'll pay depends on the size of the transaction, and if the network is congested, you have to pay a higher fee to outbid others and get your transaction confirmed fast.
Although you can always consider customizing your fee
Customizing your fee rate to a lower one in a situation of network congestion is not a solution because you will wait for days, weeks or most nodes will even purge it out of their mempool, it is best to check
https://mempool.space/ before making a transaction to choose the priority you want, either low, medium or high.