As far as I know, they just cut the next block fees, So that your transaction is mined on the next block, maybe it will take less than 10 minutes to be confirmed.
I'm aware that they do not take any cuts from the fee and spend everything based on the price, but if the average fee price will spike to a lot higher, will stake allow us to exchange our btc to some other altcoins available there to save on fees? It becomes too costly when I just have $20 to withdraw and the fee is worth more than $7.5, I will get almost nothing on my hands.
This is really a matter of thinking, Good proposal and I support you. Nowadays Bitcoin's transaction fees have really gone up a lot, there is a lot of suffering to transact small amounts.
But have you noticed that, Currently, it is not possible to withdraw 20$ if your fund remains on BTC or ETH? Maybe the minimum withdrawal amount is $30+ now, So you can't withdraw $20.
If stake gives us an option to swap our funds to other altcoins then it will be better.