Yes, pool pays fees. In your situation it might be useful to choose pool with good hashrate and low minimum payment. This link below can help you to choose:
https://miningpoolstats.stream/ethereumFrom that link, it's either Sparkpool or where I already am, Ethermine. HiveON pool has 0% fees, but the hashrate is only 17.58 TH/s compared to 77.32 on Ethermine. Is that a big factor in profits even accounting for fees?