On Ethereum blockchain, it is Ethereum aka ETH.
You pay fee in BTC or ETH but transaction fee value can be higher and more expensive if BTC and ETH becomes more expensive.
The smallest unit of Ether is Gwei, 1 Gwei = 0.000000001 ETH.
Just to be clear:
If you use BTC, you will pay fee in BTC, there's no way for you to pay in ETH.
There's no relation between other network fee with BTC and ETH network.
Assuming I hold 0.001 ETH mainly for a gas fee only and the value was 0.56$ then price surge up, is the increment going to be on the Sats? E.g ( 0.002 ETH ) or the increment is going to be on the value? E.g 0.001 ETH = 1.06$?
The increment will going to be on the Gwei, no the sats (Bitcoin) or the value.
In Ethereum network the calculation is using this formula: Total Gas Fee = Gas units (limit) x (Base fee + Tip) [1]. But I'm not really understand how to explain it, usually people will pay the fees based on the wallet calculation (low, medium, fast), not customized.
[1]
https://bitkan.com/learn/how-to-calculate-eth-gas-fees-3496