The sender can choose their own price or accept a preset amount. The recipient of transaction fees is?
It’s actually set based on the congestion on the mempool, that is why you will see different fees base on time priority. You can actually customize your own transaction fee if you use wallets that actually have that feature. Example is Electrum and BlueWallet.
They appear to be independent of the incentive for finding a block, but as this becomes more expensive, will transaction fees be the only source of funding for BTC? as so, who will profit from them?
The profit of that transaction fee is for the miners and that’s is not all they also benefit on block reward which diminishes every four years as said by bitcoin Smith above