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? I guess it doesn't hurt to switch and try it out after my next payout (at 0.1 now apparently).
When the block reward is same - it does not matter. You need some pool that catches at least 3-5 blocks at the day. Your % of pool will be bigger and your reward will be higher. With the big pools - you will get little reward for even catched block, but you will get lots of rewards.
The main problem i found - the time when we get a wave with blocks with huge rewards. In this wave big pools get lots of blocks and little pool can get zero. And no one of these reward will belong to you.