1. Software (like electrum) and hardware (like trezor and ledger) are generally the safest places for storing crypto - as long as you write down your seed and don't install shady software on your computer.
2. There are decentralised exchanges and peer to peer ones. Both generally don't hold your funds in custody, and if they do it's only for a few hours to provide escrow - some send angry emails if you leave your funds on there for too long too. Localbitcoins always felt a lot safer than most exchanges (I'd probably rate it above binance and coinbase too for safety).