1) Does Binance let you keep your keys? Or are they like a coinbase?
No. They are a full custodial exchange, so they control all coins (like Coinbase).
2) Should the safety concern in regards to your private keys being stolen/compromised be as high as it it with say a wallet like Electrum?
Ehm, that's a thuff one. If you do everything right, you will most likely never get hacked. Most (good) exchanges also focus a lot on their wallet security and store most of the coins on secure cold storage wallets, but a lot of them already got hacked, so... I personally prefer to do everything myself.
3) If they don't let you keep your keys, isn't the up side to that that your crypto is not as easy to steal?
Depends. See what I said above.
4) What safety precautions should one follow when using Binance?
Use 2FA (not SMS 2fa), don't fall for phishing websites, don't let someone hack your email, always triple check the addresses you are sending the coins to, etc... the same as any other website.
5) If they implement a 2FA when sending your crypto, doesn't that mean that your funds can never be stolen without getting the pass code they send you to verify the transaction?
No. If someone hacks them, they can change your 2FA. If a rogue employee wants to steal your coins, they can change your 2FA (or literally just steal the exchange's wallet). The only reason you need to use 2FA is to be safe against external hackers that may find out about your password.