How safe is BSC (bep20) chain? Is it okay to store a large amount of BTC and USDT on that chain? If Binance shutdown will the chain still operate normally?
Never store your funds on exchanges no matter how reliable they might seem to be.
not your keys not your coins. Exchanges are owned by centralized bodies and have full control of your funds so you don't know what might happen next.
Get your self a hardware wallets( Trezor) or rather, you can create an airgapped device and install electrum in it. Airgapped wallets are completely offline with no access to the Internet.