How safe is BSC (bep20) chain?
It's not really safe as it has been secured by a few validators but the credibility of these validators can be trusted. been using it for years and i never encounter serious problem with BSC chain.
Is it okay to store a large amount of BTC and USDT on that chain?
It's okay but if i were you and i would prefer to store it on the truly decentralized blockchains which have been secured by hundreds or thousands of validators to ensure the blockchain can always run when some validators decide to stop to run.
If Binance shutdown will the chain still operate normally?
As long as validators are not getting shut down and it can still operate normallly. It's not related to the binance. BSC was independent blockchain.